<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/5a32494f-ee17-40d1-8874-e3139abd510a/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0">I<span class="fs1 ls1 ws0">NTRODUÇÃO ÀS </span><span class="ls2">E<span class="fs1 ls5 ws1">Q U A Ç Õ E S</span></span></div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls3">D<span class="fs1 ls1 ws2">IFERENCIAIS </span><span class="ls4">O<span class="fs1 ls1 ws3">RDINÁRIAS<span class="_0 blank"></span></span></span></div><div class="t m0 x3 h3 y3 ff1 fs2 fc1 sc0 ls5 ws0">Reginaldo J. Santos</div><div class="t m0 x4 h3 y4 ff1 fs2 fc2 sc0 ls5 ws0">Departamento de Matemática-ICEx</div><div class="t m0 x5 h3 y5 ff1 fs2 fc2 sc0 ls5 ws0">Universidade Federal de Minas Gerais</div><div class="t m0 x6 h4 y6 ff2 fs2 fc1 sc0 ls5">http://www.mat.ufmg.br/\u02dcregi</div><div class="c x7 y7 w2 h5"><div class="t m0 x8 h6 y8 ff3 fs3 fc2 sc0 ls5 ws4">x<span class="fs4 v1">1</span></div><div class="t m0 x9 h6 y9 ff3 fs3 fc2 sc0 ls5 ws4">x<span class="fs4 v1">2</span></div></div><div class="t m0 x8 h3 ya ff1 fs2 fc2 sc0 ls5 ws0">Imprensa Universitária da UFMG - Belo Horizonte</div><div class="t m0 xa h3 yb ff1 fs2 fc0 sc0 ls5 ws0">Julho 2013</div><a class="l"><div class="d m1" style="border-style:none;position:absolute;left:207.704000px;bottom:323.117000px;width:94.828000px;height:12.961000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l"><div class="d m1" style="border-style:none;position:absolute;left:173.940000px;bottom:283.121000px;width:162.356000px;height:11.993000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi xb yc w3 h7" alt="" src="https://files.passeidireto.com/5a32494f-ee17-40d1-8874-e3139abd510a/bg2.png"><div class="t m0 xc h8 yd ff1 fs5 fc2 sc0 ls5 ws0">Introdução às Equações Difer<span class="_1 blank"></span>enciais Ordinárias</div><div class="t m0 xd h9 ye ff1 fs5 fc2 sc0 ls5 ws5">Copyright <span class="v2">c</span></div><div class="t m0 xe h8 ye ff4 fs5 fc2 sc0 ls6">\ue00d<span class="ff1 ls5 ws0">2013 by <span class="fc1">Reginaldo J. Santos </span>(2013.10.23)</span></div><div class="t m0 xf h8 yf ff1 fs5 fc2 sc0 ls5 ws0">Nenhuma parte desta publicação poderá ser repr<span class="_1 blank"></span>oduzida por qualquer meio sem a prévia autorização, por</div><div class="t m0 x10 h8 y10 ff1 fs5 fc2 sc0 ls5 ws0">escrito, do autor<span class="_2 blank"></span>.</div><div class="t m0 x11 h8 y11 ff1 fs5 fc2 sc0 ls5 ws0">Editor<span class="_2 blank"></span>, Coordenador de Revisão, Supervisor de Pr<span class="_1 blank"></span>odução, Capa e Ilustrações:</div><div class="t m0 x12 h8 y12 ff1 fs5 fc1 sc0 ls5 ws0">Reginaldo J. Santos</div><div class="t m0 x13 h8 y13 ff1 fs5 fc2 sc0 ls5 ws0">ISBN 978-85-7470-021-2</div><div class="t m0 x12 ha y14 ff5 fs5 fc2 sc0 ls5 ws0">Ficha Catalográ\ufb01ca</div><div class="t m0 x14 h8 y15 ff1 fs5 fc2 sc0 ls5 ws0">Santos, Reginaldo J.</div><div class="t m0 x15 h8 y16 ff1 fs5 fc2 sc0 ls5 ws0">S237i<span class="_3 blank"> </span>Introdução às Equações Difer<span class="_1 blank"></span>enciais Ordinárias / <span class="fc1">Reginaldo J. Santos</span></div><div class="t m0 x14 h8 y17 ff1 fs5 fc2 sc0 ls5 ws0">- Belo Horizonte:<span class="_4 blank"> </span>Imprensa Universitária da UFMG, 2013.</div><div class="t m0 x16 h8 y18 ff1 fs5 fc2 sc0 ls5 ws0">1.<span class="_4 blank"> </span>Equações Diferenciais<span class="_5 blank"> </span>I. Título</div><div class="t m0 x17 h8 y19 ff1 fs5 fc2 sc0 ls5 ws6">CDD: 515.3</div><a class="l"><div class="d m1" style="border-style:none;position:absolute;left:232.926000px;bottom:385.300000px;width:86.774000px;height:12.010000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l"><div class="d m1" style="border-style:none;position:absolute;left:211.731000px;bottom:316.756000px;width:86.774000px;height:12.010000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l"><div class="d m1" style="border-style:none;position:absolute;left:350.495000px;bottom:213.548000px;width:86.774000px;height:13.947000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x18 y1a w4 hb" alt="" src="https://files.passeidireto.com/5a32494f-ee17-40d1-8874-e3139abd510a/bg3.png"><div class="t m0 x18 h2 y1b ff1 fs0 fc0 sc0 ls7">S<span class="fs1 ls1 ws3">UMÁRIO<span class="_0 blank"></span></span></div><div class="t m0 x18 h3 y1c ff1 fs2 fc3 sc0 ls8">A<span class="fs6 ls9 ws7">PRESENT<span class="_2 blank"></span>AÇÃO <span class="ff6 fs2 fc2 ls5">viii</span></span></div><div class="t m0 x18 hc y1d ff1 fs2 fc3 sc0 ls5 ws8">1<span class="_6 blank"> </span>E <span class="fs6 ws9">Q U A Ç Õ E S<span class="_4 blank"> </span></span><span class="lsa">D<span class="fs6 ls9 wsa">IFERENCIAIS DE </span><span class="lsb">1</span></span><span class="fs7 v3">a</span></div><div class="t m0 x19 hd y1e ff1 fs7 fc3 sc0 lsc">.<span class="fs2 lsd v4">O<span class="fs6 ls9 wsb">RDEM <span class="ff6 fs2 fc2 ls5">1</span></span></span></div><div class="t m0 x1a h8 y1f ff6 fs5 fc3 sc0 ls5 ws0">1.1<span class="_7 blank"> </span>Introdução às Equações Diferenciais<span class="_8 blank"> </span><span class="ff1 fc2 lse wsc">....................................... </span><span class="fc2">1</span></div><div class="t m0 x1b h8 y20 ff6 fs5 fc3 sc0 ls5 wsd">1.1.1<span class="_9 blank"> </span>Classi\ufb01cação <span class="ff1 fc2 lse wse">.............................................. </span><span class="fc2">7</span></div><div class="t m0 x1b h8 y21 ff6 fs5 fc3 sc0 ls5 ws0">1.1.2<span class="_9 blank"> </span>Soluções de Equações Ordinárias<span class="_8 blank"> </span><span class="ff1 fc2 lse wsf">.................................... </span><span class="fc2">8</span></div><div class="t m0 x1b he y22 ff6 fs5 fc3 sc0 ls5 ws0">1.1.3<span class="_9 blank"> </span>Equações Ordinárias de 1<span class="fs8 v5">a</span></div><div class="t m0 x1c hf y23 ff6 fs8 fc3 sc0 lsf">.<span class="fs5 ls5 ws10 v1">Ordem <span class="ff1 fc2 lse ws11">................................... <span class="ff6 ls5">11</span></span></span></div><div class="t m0 x1b h8 y24 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws13">................................................... </span><span class="fc2">13</span></div><div class="t m0 x1a h10 y25 ff6 fs5 fc3 sc0 ls5 ws0">1.2<span class="_7 blank"> </span>Equações Lineares de 1<span class="fs8 v5">a</span></div><div class="t m0 x1d h11 y26 ff6 fs8 fc3 sc0 lsf">.<span class="fs5 ls5 ws14 v1">Ordem <span class="ff1 fc2 lse ws15">........................................ <span class="ff6 ls5">14</span></span></span></div><div class="t m0 x1b h8 y27 ff6 fs5 fc3 sc0 ls5 ws0">1.2.1<span class="_9 blank"> </span>Equações em que<span class="_4 blank"> </span><span class="ff7 ls10">p<span class="ff8 fs9 ls11">(</span><span class="ls12">t</span></span><span class="ff8 fs9 ws16">) = </span><span class="ff1 ls13">0<span class="fc2 lse ws17">...................................... </span></span><span class="fc2">14</span></div><div class="t m0 x1b h8 y28 ff6 fs5 fc3 sc0 ls5 ws0">1.2.2<span class="_9 blank"> </span>Equações Lineares - Caso Geral<span class="_a blank"> </span><span class="ff1 fc2 lse ws18">.................................... </span><span class="fc2">17</span></div><div class="t m0 x1b h12 y29 ff6 fs5 fc3 sc0 ls5 ws0">1.2.3<span class="_9 blank"> </span>Como chegar ao fator integ<span class="_1 blank"></span>rante <span class="ff9 ls14">µ<span class="ff8 fs9 ls11">(</span><span class="ff7 ls12">t</span></span><span class="ff8 fs9 ws19">) = </span><span class="ff7 ls15">e<span class="ffa fsa ls16 v6">R</span><span class="fs8 ls17 v5">p<span class="ff8 fsa ls18">(</span><span class="ls19">t<span class="ff8 fsa ls18">)</span><span class="ls5 ws1a">d t<span class="_4 blank"> </span></span></span></span></span><span class="ls1a">?<span class="ff1 fc2 lse ws1b">........................... </span></span><span class="fc2">23</span></div><div class="t m0 x1b h8 y2a ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws13">................................................... </span><span class="fc2">25</span></div><div class="t m0 x1a h8 y2b ff6 fs5 fc3 sc0 ls5 ws0">1.3<span class="_7 blank"> </span>Equações Separáveis<span class="_b blank"> </span><span class="ff1 fc2 lse ws1c">.............................................. </span><span class="fc2">27</span></div><div class="t m0 x1b h8 y2c ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws13">................................................... </span><span class="fc2">36</span></div><div class="t m0 x1a h8 y2d ff6 fs5 fc3 sc0 ls5 ws0">1.4<span class="_7 blank"> </span>Equações Exatas<span class="_c blank"> </span><span class="ff1 fc2 lse ws1d">................................................ </span><span class="fc2">38</span></div><div class="t m0 x1b h8 y2e ff6 fs5 fc3 sc0 ls5 ws0">1.4.1<span class="_9 blank"> </span>F<span class="_1 blank"></span>atores Integrantes<span class="_d blank"> </span><span class="ff1 fc2 lse ws1e">........................................... </span><span class="fc2">45</span></div><div class="t m0 x1b h8 y2f ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws13">................................................... </span><span class="fc2">49</span></div><div class="t m0 x1a h10 y30 ff6 fs5 fc3 sc0 ls5 ws0">1.5<span class="_7 blank"> </span>Substituições em Equações de 1<span class="fs8 v5">a</span></div><div class="t m0 x1e h11 y31 ff6 fs8 fc3 sc0 lsf">.<span class="fs5 ls5 ws1f v1">Ordem <span class="ff1 fc2 lse ws18">.................................... <span class="ff6 ls5">52</span></span></span></div><div class="t m0 x1b h10 y32 ff6 fs5 fc3 sc0 ls5 ws0">1.5.1<span class="_9 blank"> </span>Equações Homogêneas de 1<span class="fs8 v5">a</span></div><div class="t m0 x1f hf y33 ff6 fs8 fc3 sc0 lsf">.<span class="fs5 ls5 ws20 v1">Ordem <span class="ff1 fc2 lse ws21">.................................. <span class="ff6 ls5">52</span></span></span></div><a class="l" data-dest-detail='[8,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:13.177000px;bottom:266.028000px;width:81.901000px;height:11.647000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[11,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:13.177000px;bottom:244.092000px;width:213.243000px;height:11.696000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[11,"XYZ",14.173,214.878,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:231.799000px;width:157.716000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[17,"XYZ",14.173,287.245,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:219.840000px;width:81.992000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[18,"XYZ",14.173,182.649,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:207.881000px;width:157.508000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[21,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:195.922000px;width:159.635000px;height:11.869000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[23,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:185.976000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[24,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:172.005000px;width:144.203000px;height:11.868000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[24,"XYZ",14.173,255.219,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:159.508000px;width:138.724000px;height:12.538000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[27,"XYZ",14.173,358.41,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:148.087000px;width:152.317000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[33,"XYZ",14.173,136.216,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:134.439000px;width:223.834000px;height:14.641000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[35,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:125.031000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[37,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:111.060000px;width:104.178000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[46,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:101.113000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[48,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:87.142000px;width:88.487000px;height:11.307000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[55,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:75.183000px;width:104.178000px;height:11.308000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[59,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:65.237000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[62,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:51.266000px;width:175.077000px;height:11.868000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[62,"XYZ",14.173,324.795,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:39.307000px;width:170.425000px;height:11.869000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x18 y34 w4 h13" alt="" src="https://files.passeidireto.com/5a32494f-ee17-40d1-8874-e3139abd510a/bg4.png"><div class="t m0 x18 h14 y35 ff6 fsb fc3 sc0 ls5 ws22">iv <span class="fs5 fc0 ws23">Sumário</span></div><div class="t m0 x1b h8 yd ff6 fs5 fc3 sc0 ls5 ws0">1.5.2<span class="_9 blank"> </span>Equações de Bernoulli<span class="_e blank"> </span><span class="ff1 fc2 lse ws24">......................................... </span><span class="fc2">55</span></div><div class="t m0 x1b h8 y36 ff6 fs5 fc3 sc0 ls5 ws0">1.5.3<span class="_9 blank"> </span>Equações de Ricatti<span class="_6 blank"> </span><span class="ff1 fc2 lse ws25">.......................................... </span><span class="fc2">57</span></div><div class="t m0 x1b h15 y37 ff6 fs5 fc3 sc0 ls5 ws0">1.5.4<span class="_9 blank"> </span>Equações da forma <span class="ff7 ls1b">y<span class="ff4 fsa ls1c v5">0</span><span class="ff8 fs9 ls1d">=</span><span class="ls1e">F<span class="ff8 fs9 ls1f">(</span><span class="ls5 ws26">a x<span class="_f blank"> </span><span class="ff8 fs9 ls20">+</span><span class="ws27">by <span class="ff8 fs9 ls21">)</span><span class="ff1 fc2 lse ws28">................................. </span></span></span></span></span><span class="fc2">60</span></div><div class="t m0 x1b h8 y38 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws13">................................................... </span><span class="fc2">63</span></div><div class="t m0 x1a h8 y39 ff6 fs5 fc3 sc0 ls5 ws29">1.6<span class="_7 blank"> </span>Aplicações <span class="ff1 fc2 lse ws13">................................................... </span><span class="fc2">64</span></div><div class="t m0 x1b h8 y3a ff6 fs5 fc3 sc0 ls5 ws0">1.6.1<span class="_9 blank"> </span>Dinâmica P<span class="_1 blank"></span>opulacional<span class="_10 blank"> </span><span class="ff1 fc2 lse ws24">......................................... </span><span class="fc2">64</span></div><div class="t m0 x1b h8 y3b ff6 fs5 fc3 sc0 ls5 ws0">1.6.2<span class="_9 blank"> </span>Decaimento Radioativo<span class="_11 blank"> </span><span class="ff1 fc2 lse ws24">......................................... </span><span class="fc2">72</span></div><div class="t m0 x1b h8 y3c ff6 fs5 fc3 sc0 ls5 ws2a">1.6.3<span class="_9 blank"> </span>Misturas <span class="ff1 fc2 lse ws1d">................................................ </span><span class="fc2">76</span></div><div class="t m0 x1b h8 y3d ff6 fs5 fc3 sc0 ls5 ws0">1.6.4<span class="_9 blank"> </span>Lei de Resfriamento de Newton<span class="_11 blank"> </span><span class="ff1 fc2 lse ws2b">..................................... </span><span class="fc2">81</span></div><div class="t m0 x1b h8 y3e ff6 fs5 fc3 sc0 ls5 ws0">1.6.5<span class="_9 blank"> </span>Lei de T<span class="_2 blank"></span>orricelli<span class="_11 blank"> </span><span class="ff1 fc2 lse ws2c">............................................. </span><span class="fc2">84</span></div><div class="t m0 x1b h8 y3f ff6 fs5 fc3 sc0 ls5 ws0">1.6.6<span class="_9 blank"> </span>V<span class="_2 blank"></span>elocidade de Escape<span class="_12 blank"> </span><span class="ff1 fc2 lse ws24">......................................... </span><span class="fc2">88</span></div><div class="t m0 x1b h8 y40 ff6 fs5 fc3 sc0 ls5 ws0">1.6.7<span class="_9 blank"> </span>Resistência em Fluidos<span class="_11 blank"> </span><span class="ff1 fc2 lse ws24">......................................... </span><span class="fc2">90</span></div><div class="t m0 x1b h8 y41 ff6 fs5 fc3 sc0 ls5 ws23">1.6.8<span class="_9 blank"> </span>Circuitos<span class="_f blank"> </span>Elétricos<span class="_12 blank"> </span><span class="ff1 fc2 lse ws1e">........................................... </span><span class="fc2">95</span></div><div class="t m0 x1b h8 y42 ff6 fs5 fc3 sc0 ls5 ws2d">1.6.9<span class="_9 blank"> </span>Juros <span class="ff1 fc2 lse ws2e">.................................................. </span><span class="fc2">98</span></div><div class="t m0 x1b h10 y43 ff6 fs5 fc3 sc0 ls5 ws0">1.6.10<span class="_12 blank"> </span>Reações Químicas de 2<span class="fs8 v5">a</span></div><div class="t m0 x1e hf y44 ff6 fs8 fc3 sc0 lsf">.<span class="fs5 ls5 ws2f v1">Ordem <span class="ff1 fc2 lse ws30">.................................... <span class="ff6 ls5">108</span></span></span></div><div class="t m0 x1b h8 y45 ff6 fs5 fc3 sc0 ls5 ws23">1.6.11<span class="_12 blank"> </span>T<span class="_2 blank"></span>rajetórias<span class="_f blank"> </span>Ortogonais<span class="_6 blank"> </span><span class="ff1 fc2 lse ws31">......................................... </span><span class="fc2">115</span></div><div class="t m0 x1b h8 y46 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">119</span></div><div class="t m0 x1a h8 y47 ff6 fs5 fc3 sc0 ls5 ws0">1.7<span class="_7 blank"> </span>Análise Qualitativa<span class="_12 blank"> </span><span class="ff1 fc2 lse ws33">............................................... </span><span class="fc2">128</span></div><div class="t m0 x1b h8 y48 ff6 fs5 fc3 sc0 ls5 ws0">1.7.1<span class="_9 blank"> </span>Equações Autônomas <span class="ff1 fc2 lse ws34">.......................................... </span><span class="fc2">128</span></div><div class="t m0 x1b h8 y49 ff6 fs5 fc3 sc0 ls5 ws0">1.7.2<span class="_9 blank"> </span>Campo de Direções<span class="_f blank"> </span><span class="ff1 fc2 lse ws35">........................................... </span><span class="fc2">132</span></div><div class="t m0 x1b h8 y4a ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">133</span></div><div class="t m0 x1a h8 y4b ff6 fs5 fc3 sc0 ls5 ws0">1.8<span class="_7 blank"> </span>Existência e Unicidade de Soluções<span class="_13 blank"> </span><span class="ff1 fc2 lse ws36">....................................... </span><span class="fc2">135</span></div><div class="t m0 x1b h8 y4c ff6 fs5 fc3 sc0 ls5 ws0">1.8.1<span class="_9 blank"> </span>Demonstração do T<span class="_14 blank"></span>eorema de Existência e Unicidade<span class="_10 blank"> </span><span class="ff1 fc2 lse ws37">.......................... </span><span class="fc2">140</span></div><div class="t m0 x1b h8 y4d ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">145</span></div><div class="t m0 x1a h8 y4e ff6 fs5 fc3 sc0 ls5 ws0">1.9<span class="_7 blank"> </span>Respostas dos Exercícios<span class="_11 blank"> </span><span class="ff1 fc2 lse ws38">............................................ </span><span class="fc2">147</span></div><div class="t m0 x18 hc y4f ff1 fs2 fc3 sc0 ls5 ws8">2<span class="_6 blank"> </span>E <span class="fs6 ws9">Q U A Ç Õ E S<span class="_4 blank"> </span></span><span class="lsa">D<span class="fs6 ls9 ws39">IFERENCIAIS </span><span class="ls22">L<span class="fs6 ls9 ws3a">INEARES DE </span><span class="ls23">2</span></span></span><span class="fs7 v3">a</span></div><div class="t m0 x20 hd y50 ff1 fs7 fc3 sc0 ls24">.<span class="fs2 lsd v4">O<span class="fs6 ls9 ws3b">RDEM <span class="ff6 fs2 fc2 ls5">258</span></span></span></div><div class="t m0 x1a h8 y51 ff6 fs5 fc3 sc0 ls5 ws0">2.1<span class="_7 blank"> </span>Equações Homogêneas - P<span class="_1 blank"></span>ar<span class="_15 blank"> </span>te I<span class="_4 blank"> </span><span class="ff1 fc2 lse ws31">......................................... </span><span class="fc2">259</span></div><div class="t m0 x1b h8 y52 ff6 fs5 fc3 sc0 ls5 ws0">2.1.1<span class="_9 blank"> </span>Soluções Fundamentais<span class="_6 blank"> </span><span class="ff1 fc2 lse ws3c">........................................ </span><span class="fc2">260</span></div><div class="t m0 x1b h8 y53 ff6 fs5 fc3 sc0 ls5 ws0">2.1.2<span class="_9 blank"> </span>Fórmula de Euler<span class="_b blank"> </span><span class="ff1 fc2 lse ws38">............................................ </span><span class="fc2">270</span></div><div class="t m0 x1b h8 y32 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">272</span></div><div class="t m0 x21 h8 y54 ff6 fs5 fc0 sc0 ls5 ws0">Introdução às Equações Diferenciais Ordinárias<span class="_16 blank"> </span><span class="ff1 fc4 ws3d">GoBack GoForward<span class="_17 blank"> </span></span>Julho 2013</div><a class="l" data-dest-detail='[3,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:13.177000px;bottom:427.878000px;width:9.070000px;height:14.097000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[65,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:397.848000px;width:115.824000px;height:11.308000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[67,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:385.836000px;width:106.529000px;height:11.307000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[70,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:373.286000px;width:176.193000px;height:12.740000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[73,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:363.824000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[74,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:349.800000px;width:64.408000px;height:11.307000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[74,"XYZ",14.173,385.062,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:337.907000px;width:116.562000px;height:11.417000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[82,"XYZ",14.173,312.303,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:327.788000px;width:118.126000px;height:9.295000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[86,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:315.776000px;width:64.667000px;height:9.295000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[91,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:303.764000px;width:148.272000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[94,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:291.752000px;width:88.407000px;height:9.295000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[98,"XYZ",14.173,208.188,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:277.847000px;width:114.519000px;height:11.188000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[100,"XYZ",14.173,208.824,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:267.727000px;width:118.315000px;height:9.525000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[105,"XYZ",14.173,116.978,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:255.715000px;width:99.356000px;height:9.525000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[108,"XYZ",14.173,136.823,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:243.703000px;width:53.699000px;height:9.295000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[118,"XYZ",14.173,174.919,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:229.679000px;width:152.243000px;height:11.868000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[125,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:217.667000px;width:114.101000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[129,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:207.667000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[138,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:195.301000px;width:92.362000px;height:9.878000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[138,"XYZ",14.173,387.54,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:181.630000px;width:113.563000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[142,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:169.618000px;width:106.061000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[143,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:159.619000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[145,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:145.594000px;width:154.777000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[150,"XYZ",14.173,263.472,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:133.582000px;width:228.540000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[155,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:123.582000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[157,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:109.677000px;width:118.654000px;height:11.417000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[268,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:13.177000px;bottom:87.690000px;width:265.017000px;height:11.696000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[269,"XYZ",14.173,173.766,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:75.343000px;width:143.440000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[270,"XYZ",14.173,108.83,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:63.331000px;width:121.513000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[280,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:53.331000px;width:96.666000px;height:9.525000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[282,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:41.409000px;width:39.432000px;height:9.434000px;background-color:rgba(255,255,255,0.000001);"></div></a><div class="d m1" style="border-style:none;position:absolute;left:210.159000px;bottom:12.736000px;width:36.064000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div><div class="d m1" style="border-style:none;position:absolute;left:249.488000px;bottom:12.736000px;width:53.080000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x18 y34 w4 h13" alt="" src="https://files.passeidireto.com/5a32494f-ee17-40d1-8874-e3139abd510a/bg5.png"><div class="t m0 x18 h14 y55 ff6 fs5 fc0 sc0 ls5 ws23">Sumário<span class="_18 blank"> </span><span class="fsb fc3">v</span></div><div class="t m0 x1a h8 yd ff6 fs5 fc3 sc0 ls5 ws0">2.2<span class="_7 blank"> </span>Equações Homogêneas - P<span class="_1 blank"></span>ar<span class="_15 blank"> </span>te II<span class="_19 blank"> </span><span class="ff1 fc2 lse ws3e">........................................ </span><span class="fc2">276</span></div><div class="t m0 x1b h8 y36 ff6 fs5 fc3 sc0 ls5 ws0">2.2.1<span class="_9 blank"> </span>Obtendo-se uma Segunda Solução (Redução de Ordem)<span class="_f blank"> </span><span class="ff1 fc2 lse ws3f">......................... </span><span class="fc2">276</span></div><div class="t m0 x1b h8 y37 ff6 fs5 fc3 sc0 ls5 ws0">2.2.2<span class="_9 blank"> </span>Equações Homogêneas com Coe\ufb01cientes Constantes<span class="_1a blank"> </span><span class="ff1 fc2 lse ws40">.......................... </span><span class="fc2">280</span></div><div class="t m0 x1b h8 y38 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">292</span></div><div class="t m0 x1a h8 y39 ff6 fs5 fc3 sc0 ls5 ws0">2.3<span class="_7 blank"> </span>Equações Não Homogêneas<span class="_6 blank"> </span><span class="ff1 fc2 lse ws34">.......................................... </span><span class="fc2">294</span></div><div class="t m0 x1b h8 y3a ff6 fs5 fc3 sc0 ls5 ws0">2.3.1<span class="_9 blank"> </span>Método de V<span class="_2 blank"></span>ar<span class="_15 blank"> </span>iação dos P<span class="_1 blank"></span>arâmetros<span class="_6 blank"> </span><span class="ff1 fc2 lse ws41">.................................. </span><span class="fc2">298</span></div><div class="t m0 x1b h8 y3b ff6 fs5 fc3 sc0 ls5 ws0">2.3.2<span class="_9 blank"> </span>Método dos Coe\ufb01cientes a Determinar para Equações com Coe\ufb01cientes Constantes<span class="_a blank"> </span><span class="ff1 fc2 lse ws42">........... </span><span class="fc2">304</span></div><div class="t m0 x1b h8 y3c ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">315</span></div><div class="t m0 x1a h8 y3d ff6 fs5 fc3 sc0 ls5 ws0">2.4<span class="_7 blank"> </span>Oscilações Livres<span class="_11 blank"> </span><span class="ff1 fc2 lse ws43">................................................ </span><span class="fc2">316</span></div><div class="t m0 x1b h8 y3e ff6 fs5 fc3 sc0 ls5 ws44">2.4.1<span class="_9 blank"> </span>Sem<span class="_f blank"> </span>Amortecimento<span class="_e blank"> </span><span class="ff1 fc2 lse ws34">.......................................... </span><span class="fc2">319</span></div><div class="t m0 x1b h8 y3f ff6 fs5 fc3 sc0 ls5 ws44">2.4.2<span class="_9 blank"> </span>Com<span class="_f blank"> </span>Amortecimento<span class="_1b blank"> </span><span class="ff1 fc2 lse ws34">.......................................... </span><span class="fc2">324</span></div><div class="t m0 x1b h8 y40 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">334</span></div><div class="t m0 x1a h8 y41 ff6 fs5 fc3 sc0 ls5 ws0">2.5<span class="_7 blank"> </span>Oscilações Forçadas<span class="_e blank"> </span><span class="ff1 fc2 lse ws45">.............................................. </span><span class="fc2">336</span></div><div class="t m0 x1b h8 y42 ff6 fs5 fc3 sc0 ls5 ws44">2.5.1<span class="_9 blank"> </span>Sem<span class="_f blank"> </span>Amortecimento<span class="_e blank"> </span><span class="ff1 fc2 lse ws34">.......................................... </span><span class="fc2">336</span></div><div class="t m0 x1b h8 y43 ff6 fs5 fc3 sc0 ls5 ws44">2.5.2<span class="_9 blank"> </span>Com<span class="_f blank"> </span>Amortecimento<span class="_1b blank"> </span><span class="ff1 fc2 lse ws34">.......................................... </span><span class="fc2">343</span></div><div class="t m0 x1b h8 y45 ff6 fs5 fc3 sc0 ls5 ws23">2.5.3<span class="_9 blank"> </span>Circuitos<span class="_f blank"> </span>Elétricos<span class="_12 blank"> </span><span class="ff1 fc2 lse ws35">........................................... </span><span class="fc2">349</span></div><div class="t m0 x1b h8 y46 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">352</span></div><div class="t m0 x1a h8 y47 ff6 fs5 fc3 sc0 ls5 ws0">2.6<span class="_7 blank"> </span>Soluções em Séries de P<span class="_1 blank"></span>otências<span class="_1c blank"> </span><span class="ff1 fc2 lse ws3c">........................................ </span><span class="fc2">355</span></div><div class="t m0 x1b h8 y48 ff6 fs5 fc3 sc0 ls5 ws0">2.6.1<span class="_9 blank"> </span>Demonstração do T<span class="_14 blank"></span>eorema de Existência de Soluções em Séries<span class="_8 blank"> </span><span class="ff1 fc2 lse ws46">..................... </span><span class="fc2">371</span></div><div class="t m0 x1b h8 y49 ff6 fs5 fc3 sc0 ls5 ws0">2.6.2<span class="_9 blank"> </span>Demonstração das Propriedades de Séries de Potências<span class="_8 blank"> </span><span class="ff1 fc2 lse ws47">......................... </span><span class="fc2">377</span></div><div class="t m0 x1b h8 y4a ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">380</span></div><div class="t m0 x1a h8 y4b ff6 fs5 fc3 sc0 ls5 ws0">2.7<span class="_7 blank"> </span>Mudanças de V<span class="_2 blank"></span>ar<span class="_15 blank"> </span>iáv<span class="_1 blank"></span>eis<span class="_1c blank"> </span><span class="ff1 fc2 lse ws48">............................................. </span><span class="fc2">387</span></div><div class="t m0 x1b h8 y4c ff6 fs5 fc3 sc0 ls5 ws0">2.7.1<span class="_9 blank"> </span>Equações que não Contém <span class="ff7 ls25">y<span class="ff1 fc2 lse ws49">...................................... </span></span><span class="fc2">387</span></div><div class="t m0 x1b h8 y4d ff6 fs5 fc3 sc0 ls5 ws0">2.7.2<span class="_9 blank"> </span>Equações que não Contém <span class="ff7 ls26">t<span class="ff1 fc2 lse ws49">...................................... </span></span><span class="fc2">388</span></div><div class="t m0 x1b h8 y4e ff6 fs5 fc3 sc0 ls5 ws0">2.7.3<span class="_9 blank"> </span>Equações de Euler<span class="_c blank"> </span><span class="ff1 fc2 lse ws35">........................................... </span><span class="fc2">390</span></div><div class="t m0 x1b h8 y56 ff6 fs5 fc3 sc0 ls5 ws0">2.7.4<span class="_9 blank"> </span>Outras Mudanças<span class="_1d blank"> </span><span class="ff1 fc2 lse ws38">............................................ </span><span class="fc2">392</span></div><div class="t m0 x1b h8 y57 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">394</span></div><div class="t m0 x1a h8 y58 ff6 fs5 fc3 sc0 ls5 ws0">2.8<span class="_7 blank"> </span>Respostas dos Exercícios<span class="_11 blank"> </span><span class="ff1 fc2 lse ws38">............................................ </span><span class="fc2">395</span></div><div class="t m0 x18 h3 y53 ff1 fs2 fc3 sc0 ls5 ws4a">3<span class="_6 blank"> </span>T <span class="fs6 ws9">R A N S F O R M A D A<span class="_4 blank"> </span>D E<span class="_8 blank"> </span></span><span class="ls22">L<span class="fs6 ls9 ws4b">APLACE </span></span><span class="ff6 fc2">473</span></div><div class="t m0 x1a h8 y32 ff6 fs5 fc3 sc0 ls5 ws4c">3.1<span class="_7 blank"> </span>Introdução <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">473</span></div><div class="t m0 x18 h8 y54 ff6 fs5 fc0 sc0 ls5 ws0">Julho 2013<span class="_1e blank"> </span><span class="ff1 fc4 ws3d">GoBack GoForward<span class="_1f blank"> </span></span><span class="fc1">Reginaldo J. Santos</span></div><a class="l" data-dest-detail='[3,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:490.165000px;bottom:428.027000px;width:6.894000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[286,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:397.848000px;width:145.712000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[286,"XYZ",14.173,374.049,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:385.836000px;width:240.495000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[290,"XYZ",14.173,212.05,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:373.824000px;width:229.586000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[302,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:363.824000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[304,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:349.800000px;width:129.353000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[308,"XYZ",14.173,238.891,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:337.788000px;width:166.165000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[314,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:325.775000px;width:339.153000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[325,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:315.776000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[326,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:301.751000px;width:88.916000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[329,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:291.752000px;width:108.203000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[334,"XYZ",14.173,118.84,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:279.740000px;width:108.651000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[344,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:267.727000px;width:39.432000px;height:9.525000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[346,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:253.703000px;width:100.940000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[346,"XYZ",14.173,314.667,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:243.703000px;width:108.203000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[353,"XYZ",14.173,156.318,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:231.691000px;width:108.651000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[359,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:219.679000px;width:99.356000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[362,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:207.667000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[365,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:193.642000px;width:146.778000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[381,"XYZ",14.173,157.985,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:181.630000px;width:269.526000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[387,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:169.618000px;width:239.349000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[390,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:159.619000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[397,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:145.594000px;width:109.179000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[397,"XYZ",14.173,384.666,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:133.044000px;width:140.382000px;height:12.075000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[398,"XYZ",14.173,282.723,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:121.570000px;width:138.718000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[400,"XYZ",14.173,296.813,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:109.558000px;width:102.903000px;height:11.307000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[402,"XYZ",14.173,229.756,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:97.546000px;width:98.281000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[404,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:87.546000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[405,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:73.641000px;width:118.654000px;height:11.417000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[483,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:13.177000px;bottom:53.312000px;width:165.473000px;height:9.989000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[483,"XYZ",14.173,217.174,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:39.307000px;width:63.511000px;height:11.307000px;background-color:rgba(255,255,255,0.000001);"></div></a><div class="d m1" style="border-style:none;position:absolute;left:210.159000px;bottom:12.736000px;width:36.064000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div><div class="d m1" style="border-style:none;position:absolute;left:249.488000px;bottom:12.736000px;width:53.080000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div><a class="l"><div class="d m1" style="border-style:none;position:absolute;left:422.191000px;bottom:12.736000px;width:74.868000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x18 y34 w4 h13" alt="" src="https://files.passeidireto.com/5a32494f-ee17-40d1-8874-e3139abd510a/bg6.png"><div class="t m0 x18 h14 y35 ff6 fsb fc3 sc0 ls5 ws22">vi <span class="fs5 fc0 ws23">Sumário</span></div><div class="t m0 x1b h8 yd ff6 fs5 fc3 sc0 ls5 ws0">3.1.1<span class="_9 blank"> </span>Demonstração da Injetividade da T<span class="_14 blank"></span>ransf<span class="_1 blank"></span>or<span class="_15 blank"> </span>mada de Laplace<span class="_1d blank"> </span><span class="ff1 fc2 lse ws4d">........................ </span><span class="fc2">486</span></div><div class="t m0 x1b h8 y36 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">490</span></div><div class="t m0 x1a h8 y37 ff6 fs5 fc3 sc0 ls5 ws0">3.2<span class="_7 blank"> </span>Problemas de V<span class="_2 blank"></span>alor Inicial<span class="_11 blank"> </span><span class="ff1 fc2 lse ws38">............................................ </span><span class="fc2">492</span></div><div class="t m0 x1b h8 y38 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">503</span></div><div class="t m0 x1a h8 y39 ff6 fs5 fc3 sc0 ls5 ws0">3.3<span class="_7 blank"> </span>Função de Heaviside e Equações com T<span class="_0 blank"></span>er<span class="_15 blank"> </span>mo Não Homogêneo Descontínuo<span class="_12 blank"> </span><span class="ff1 fc2 lse ws4e">................... </span><span class="fc2">505</span></div><div class="t m0 x1b h8 y3a ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">522</span></div><div class="t m0 x1a h8 y3b ff6 fs5 fc3 sc0 ls5 ws0">3.4<span class="_7 blank"> </span>T<span class="_14 blank"></span>ransformada de Laplace do Delta de Dirac<span class="_12 blank"> </span><span class="ff1 fc2 lse ws4f">................................... </span><span class="fc2">525</span></div><div class="t m0 x1b h8 y3c ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">532</span></div><div class="t m0 x1a h8 y3d ff6 fs5 fc3 sc0 ls5 ws50">3.5<span class="_7 blank"> </span>Conv<span class="_1 blank"></span>olução <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">533</span></div><div class="t m0 x1b h8 y3e ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">543</span></div><div class="t m0 x1a h8 y3f ff6 fs5 fc3 sc0 ls5 ws0">3.6<span class="_7 blank"> </span>T<span class="_14 blank"></span>abela de T<span class="_14 blank"></span>ransformadas de Laplace<span class="_1d blank"> </span><span class="ff1 fc2 lse ws36">....................................... </span><span class="fc2">544</span></div><div class="t m0 x1a h8 y40 ff6 fs5 fc3 sc0 ls5 ws0">3.7<span class="_7 blank"> </span>Respostas dos Exercícios<span class="_11 blank"> </span><span class="ff1 fc2 lse ws38">............................................ </span><span class="fc2">545</span></div><div class="t m0 x18 h3 y59 ff1 fs2 fc3 sc0 ls5 ws51">4<span class="_6 blank"> </span>S <span class="fs6 ls9 ws52">ISTEMAS DE </span><span class="ls27">E<span class="fs6 ls9 ws53">QUAÇÕES </span><span class="lsa">D<span class="fs6 ls9 ws39">IFERENCIAIS </span><span class="ls22">L<span class="fs6 ls9 ws54">INEARES </span></span></span></span><span class="ff6 fc2">594</span></div><div class="t m0 x1a h8 y5a ff6 fs5 fc3 sc0 ls5 ws23">4.1<span class="_7 blank"> </span>A<span class="_f blank"> </span>Matriz<span class="_1d blank"> </span><span class="ff7 ls28">A</span><span class="ws0">é Diagonalizável em <span class="ffb fs9 ls29">R</span><span class="ff1 fc2 lse ws3e">........................................ </span><span class="fc2">602</span></span></div><div class="t m0 x1b h8 y5b ff6 fs5 fc3 sc0 ls5 ws55">4.1.1<span class="_9 blank"> </span>Sistema com <span class="ff1 ls2a">2</span><span class="ws56">Equações e <span class="ff1 ls2a">2</span><span class="ws57">Incógnitas <span class="ff1 fc2 lse ws58">................................. </span><span class="fc2">602</span></span></span></div><div class="t m0 x1b h8 y5c ff6 fs5 fc3 sc0 ls5 ws0">4.1.2<span class="_9 blank"> </span>Sistema com <span class="ff7 ls2b">n</span>Equações e <span class="ff7 ls2c">n</span><span class="ws59">Incógnitas <span class="ff1 fc2 lse ws5a">................................ </span><span class="fc2">604</span></span></div><div class="t m0 x1b h8 y5d ff6 fs5 fc3 sc0 ls5 ws0">4.1.3<span class="_9 blank"> </span>Como Encontrar as Matrizes<span class="_f blank"> </span><span class="ff7 ls2d">P</span><span class="ls2e">e<span class="ff7 ls2f">D<span class="ff1 fc2 lse ws4f">................................... </span></span></span><span class="fc2">607</span></div><div class="t m0 x1b h8 y5e ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">621</span></div><div class="t m0 x1a h8 y5f ff6 fs5 fc3 sc0 ls5 ws23">4.2<span class="_7 blank"> </span>A<span class="_f blank"> </span>Matriz<span class="_1d blank"> </span><span class="ff7 ls28">A</span><span class="ws0">é Diagonalizável em <span class="ffb fs9 ls30">C</span><span class="ff1 fc2 lse ws3c">........................................ </span><span class="fc2">625</span></span></div><div class="t m0 x1b h8 y60 ff6 fs5 fc3 sc0 ls5 ws55">4.2.1<span class="_9 blank"> </span>Sistema com <span class="ff1 ls2a">2</span><span class="ws56">Equações e <span class="ff1 ls2a">2</span><span class="ws5b">Incógnitas <span class="ff1 fc2 lse ws5c">................................. </span><span class="fc2">625</span></span></span></div><div class="t m0 x1b h8 y61 ff6 fs5 fc3 sc0 ls5 ws0">4.2.2<span class="_9 blank"> </span>Sistema com <span class="ff7 ls2b">n</span>Equações e <span class="ff7 ls2b">n</span><span class="ws5d">Incógnitas <span class="ff1 fc2 lse ws5a">................................ </span><span class="fc2">628</span></span></div><div class="t m0 x1b h8 y62 ff6 fs5 fc3 sc0 ls5 ws0">4.2.3<span class="_9 blank"> </span>Como Encontrar as Matrizes<span class="_f blank"> </span><span class="ff7 ls2d">P</span><span class="ls2e">e<span class="ff7 ls2f">D<span class="ff1 fc2 lse ws4f">................................... </span></span></span><span class="fc2">630</span></div><div class="t m0 x1b h8 y63 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">640</span></div><div class="t m0 x1a h8 y64 ff6 fs5 fc3 sc0 ls5 ws23">4.3<span class="_7 blank"> </span>A<span class="_f blank"> </span>Matriz<span class="_1d blank"> </span><span class="ff7 ls28">A</span><span class="ws0">não é Diagonalizável<span class="_4 blank"> </span><span class="ff1 fc2 lse ws31">......................................... </span><span class="fc2">642</span></span></div><div class="t m0 x1b h8 y65 ff6 fs5 fc3 sc0 ls5 ws55">4.3.1<span class="_9 blank"> </span>Sistema com <span class="ff1 ls2a">2</span><span class="ws56">Equações e <span class="ff1 ls2a">2</span><span class="ws5b">Incógnitas <span class="ff1 fc2 lse ws5c">................................. </span><span class="fc2">642</span></span></span></div><div class="t m0 x1b h8 y4f ff6 fs5 fc3 sc0 ls5 ws0">4.3.2<span class="_9 blank"> </span>Sistema com <span class="ff7 ls2b">n</span>Equações e <span class="ff7 ls2b">n</span><span class="ws5d">Incógnitas <span class="ff1 fc2 lse ws5a">................................ </span><span class="fc2">644</span></span></div><div class="t m0 x1b h8 y51 ff6 fs5 fc3 sc0 ls5 ws0">4.3.3<span class="_9 blank"> </span>Como Encontrar as Matrizes<span class="_f blank"> </span><span class="ff7 ls2d">P</span><span class="ls31">e<span class="ff7 ls32">J<span class="ff1 fc2 lse ws4f">................................... </span></span></span><span class="fc2">646</span></div><div class="t m0 x1b h8 y52 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">655</span></div><div class="t m0 x1a h8 y53 ff6 fs5 fc3 sc0 ls5 ws0">4.4<span class="_7 blank"> </span>Sistemas Não-Homogêneos (opcional)<span class="_8 blank"> </span><span class="ff1 fc2 lse ws49">...................................... </span><span class="fc2">656</span></div><div class="t m0 x1b h8 y32 ff6 fs5 fc3 sc0 ls5 ws23">4.4.1<span class="_9 blank"> </span>A<span class="_f blank"> </span>Matriz<span class="_1d blank"> </span><span class="ff7 ls28">A</span><span class="ws0">é Diagonalizável em <span class="ffb fs9 ls33">R</span><span class="ff1 fc2 lse ws4f">................................... </span><span class="fc2">657</span></span></div><div class="t m0 x21 h8 y54 ff6 fs5 fc0 sc0 ls5 ws0">Introdução às Equações Diferenciais Ordinárias<span class="_16 blank"> </span><span class="ff1 fc4 ws3d">GoBack GoForward<span class="_17 blank"> </span></span>Julho 2013</div><a class="l" data-dest-detail='[3,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:13.177000px;bottom:427.878000px;width:9.070000px;height:14.097000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[496,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:397.848000px;width:247.528000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[500,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:387.848000px;width:39.432000px;height:9.525000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[502,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:375.836000px;width:119.092000px;height:9.295000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[513,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:363.824000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[515,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:349.800000px;width:301.834000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[532,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:339.800000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[535,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:325.895000px;width:182.244000px;height:11.417000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[542,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:315.776000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[543,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:301.751000px;width:67.696000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[553,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:291.752000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[554,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:277.847000px;width:158.653000px;height:11.417000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[555,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:265.835000px;width:118.654000px;height:11.417000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[604,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:13.177000px;bottom:243.847000px;width:262.246000px;height:11.647000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[612,"XYZ",14.173,222.489,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:231.500000px;width:150.589000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[612,"XYZ",14.173,189.684,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:219.488000px;width:180.521000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[614,"XYZ",14.173,85.731,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:207.476000px;width:182.135000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[617,"XYZ",14.173,401.061,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:197.477000px;width:163.515000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[631,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:185.464000px;width:39.432000px;height:9.525000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[635,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:171.440000px;width:150.589000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[635,"XYZ",14.173,384.666,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:159.428000px;width:180.521000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[638,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:147.416000px;width:182.135000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[640,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:137.416000px;width:163.515000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[650,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:125.404000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[652,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:111.379000px;width:143.101000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[652,"XYZ",14.173,384.666,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:99.367000px;width:180.521000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[654,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:87.355000px;width:182.135000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[656,"XYZ",14.173,359.595,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:75.507000px;width:159.729000px;height:11.373000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[665,"XYZ",14.173,418.405,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:65.343000px;width:39.432000px;height:9.525000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[666,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:51.319000px;width:165.198000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[667,"XYZ",14.173,190.221,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:39.307000px;width:159.555000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><div class="d m1" style="border-style:none;position:absolute;left:210.159000px;bottom:12.736000px;width:36.064000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div><div class="d m1" style="border-style:none;position:absolute;left:249.488000px;bottom:12.736000px;width:53.080000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x18 y34 w4 h13" alt="" src="https://files.passeidireto.com/5a32494f-ee17-40d1-8874-e3139abd510a/bg7.png"><div class="t m0 x18 h14 y35 ff6 fs5 fc0 sc0 ls5 ws23">Sumário<span class="_20 blank"> </span><span class="fsb fc3">vii</span></div><div class="t m0 x1b h8 yd ff6 fs5 fc3 sc0 ls5 ws23">4.4.2<span class="_9 blank"> </span>A<span class="_f blank"> </span>Matriz<span class="_1d blank"> </span><span class="ff7 ls28">A</span><span class="ws0">é Diagonalizável em <span class="ffb fs9 ls34">C</span><span class="ff1 fc2 lse ws4f">................................... </span><span class="fc2">661</span></span></div><div class="t m0 x1b h8 ye ff6 fs5 fc3 sc0 ls5 ws23">4.4.3<span class="_9 blank"> </span>A<span class="_f blank"> </span>Matriz<span class="_1d blank"> </span><span class="ff7 ls28">A</span><span class="ws0">não é Diagonalizável<span class="_21 blank"> </span><span class="ff1 fc2 lse ws30">.................................... </span><span class="fc2">665</span></span></div><div class="t m0 x1b h8 y66 ff6 fs5 fc3 sc0 ls5 ws0">4.4.4<span class="_9 blank"> </span>Usando a T<span class="_14 blank"></span>ransformada de Laplace<span class="_11 blank"> </span><span class="ff1 fc2 lse ws4f">................................... </span><span class="fc2">669</span></div><div class="t m0 x1b h8 y67 ff6 fs5 fc3 sc0 ls5 ws0">4.4.5<span class="_9 blank"> </span>Demonstração do T<span class="_14 blank"></span>eorema de Existência e Unicidade<span class="_10 blank"> </span><span class="ff1 fc2 lse ws40">.......................... </span><span class="fc2">673</span></div><div class="t m0 x1b h8 y68 ff6 fs5 fc3 sc0 ls5 ws12">Exercícios <span class="ff1 fc2 lse ws32">................................................... </span><span class="fc2">677</span></div><div class="t m0 x1a h8 y69 ff6 fs5 fc3 sc0 ls5 ws0">4.5<span class="_7 blank"> </span>Respostas dos Exercícios<span class="_11 blank"> </span><span class="ff1 fc2 lse ws38">............................................ </span><span class="fc2">679</span></div><div class="t m0 x18 h3 y6a ff1 fs2 fc3 sc0 ls22">B<span class="fs6 ls9 ws5e">IBLIOGRAF<span class="_2 blank"></span>IA <span class="ff6 fs2 fc2 ls5">732</span></span></div><div class="t m0 x18 h3 y6b ff1 fs2 fc3 sc0 ls35">Í<span class="fs6 ls5 ws5f">N D I C E<span class="_4 blank"> </span></span><span class="ls36">A<span class="fs6 ls9 ws60">LF<span class="_1 blank"></span>ABÉTICO <span class="ff6 fs2 fc2 ls5">734</span></span></span></div><div class="t m0 x18 h8 y54 ff6 fs5 fc0 sc0 ls5 ws0">Julho 2013<span class="_1e blank"> </span><span class="ff1 fc4 ws3d">GoBack GoForward<span class="_1f blank"> </span></span><span class="fc1">Reginaldo J. Santos</span></div><a class="l" data-dest-detail='[3,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:485.814000px;bottom:427.878000px;width:11.245000px;height:14.097000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[671,"XYZ",14.173,125.824,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:397.848000px;width:159.555000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[675,"XYZ",14.173,64.04,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:385.893000px;width:152.068000px;height:11.537000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[679,"XYZ",14.173,129.444,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:374.057000px;width:163.136000px;height:11.417000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[683,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:361.983000px;width:228.540000px;height:11.536000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[687,"XYZ",14.173,366.237,null]'><div class="d m1" style="border-style:none;position:absolute;left:51.035000px;bottom:352.040000px;width:39.432000px;height:9.524000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[689,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:28.121000px;bottom:338.192000px;width:118.654000px;height:11.417000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[742,"XYZ",14.173,285.469,null]'><div class="d m1" style="border-style:none;position:absolute;left:13.177000px;bottom:318.147000px;width:71.853000px;height:9.989000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[744,"XYZ",14.173,292.439,null]'><div class="d m1" style="border-style:none;position:absolute;left:13.177000px;bottom:296.230000px;width:102.322000px;height:12.154000px;background-color:rgba(255,255,255,0.000001);"></div></a><div class="d m1" style="border-style:none;position:absolute;left:210.159000px;bottom:12.736000px;width:36.064000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div><div class="d m1" style="border-style:none;position:absolute;left:249.488000px;bottom:12.736000px;width:53.080000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div><a class="l"><div class="d m1" style="border-style:none;position:absolute;left:422.191000px;bottom:12.736000px;width:74.868000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf8" class="pf w0 h0" data-page-no="8"><div class="pc pc8 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x18 y6c w4 h16" alt="" src="https://files.passeidireto.com/5a32494f-ee17-40d1-8874-e3139abd510a/bg8.png"><div class="t m0 x18 h2 y6d ff1 fs0 fc0 sc0 ls37">A<span class="fs1 ls1 ws3">PRESENT<span class="_0 blank"></span>AÇÃO<span class="_0 blank"></span></span></div><div class="t m0 x1a h8 y6e ff1 fs5 fc2 sc0 ls5 ws61">Esse é um texto alternativo ao excelente livro Boyce-DiPrima [<span class="fc5">2</span>] para a parte de equações difer<span class="_1 blank"></span>enciais or-</div><div class="t m0 x18 h8 y6f ff1 fs5 fc2 sc0 ls5 ws62">dinárias,<span class="_22 blank"> </span>sendo mais objetivo e mais elementar<span class="_2 blank"></span>.<span class="_13 blank"> </span>Entr<span class="_1 blank"></span>etanto aqui estão apresentadas pr<span class="_1 blank"></span>ovas elementares de</div><div class="t m0 x18 h8 y70 ff1 fs5 fc2 sc0 ls5 ws63">resultados como os teor<span class="_1 blank"></span>emas de existência e unicidade para equações diferenciais e para sistemas de equações</div><div class="t m0 x18 h17 y71 ff1 fs5 fc2 sc0 ls5 ws64">diferenciais, o teorema sobr<span class="_1 blank"></span>e a existência de soluções em série de potências para equações linear<span class="_1 blank"></span>es de 2<span class="fs8 v5">a</span></div><div class="t m0 x22 h18 y72 ff1 fs8 fc2 sc0 ls38">.<span class="fs5 ls5 ws65 v4">or-</span></div><div class="t m0 x18 h8 y73 ff1 fs5 fc2 sc0 ls5 ws66">dem, a injetividade da transformada de Laplace e outros.<span class="_4 blank"> </span>O conteúdo corresponde ao programa da disciplina</div><div class="t m0 x18 h8 y74 ff1 fs5 fc2 sc0 ls5 ws66">\u2019Equações Diferenciais A<span class="_14 blank"></span>\u2019 que é ministrado para os alunos da área de ciências exatas na Universidade Federal</div><div class="t m0 x18 h8 y75 ff1 fs5 fc2 sc0 ls5 ws0">de Minas Gerais.</div><div class="t m0 x1a h17 y76 ff1 fs5 fc2 sc0 ls5 ws67">O texto é dividido em quatro capítulos.<span class="_11 blank"> </span>No Capítulo 1 apesar do título ser \u2019Equações Diferenciais de 1<span class="fs8 v5">a</span></div><div class="t m0 x23 h19 y77 ff1 fs8 fc2 sc0 ls5">.</div><div class="t m0 x18 h17 y78 ff1 fs5 fc2 sc0 ls5 ws68">Ordem\u2019 é feita uma intr<span class="_1 blank"></span>odução às equações diferenciais em geral e entr<span class="_1 blank"></span>e as equações de 1<span class="fs8 v5">a</span></div><div class="t m0 x24 h1a y79 ff1 fs8 fc2 sc0 ls39">.<span class="fs5 ls5 ws68 v4">ordem são estudadas</span></div><div class="t m0 x18 h17 y7a ff1 fs5 fc2 sc0 ls5 ws61">as equações lineares, as separáveis e as exatas.<span class="_d blank"> </span>T<span class="_14 blank"></span>em uma seção sobre substituições em equações de 1<span class="fs8 v5">a</span></div><div class="t m0 x25 h18 y7b ff1 fs8 fc2 sc0 ls3a">.<span class="fs5 ls5 ws65 v4">ordem</span></div><div class="t m0 x18 h8 y7c ff1 fs5 fc2 sc0 ls5 ws69">onde são estudadas entre outras, as equações homogêneas,<span class="_d blank"> </span>as de Bernoulli e as de Ricatti.<span class="_13 blank"> </span>T<span class="_2 blank"></span>erminamos o</div><div class="t m0 x18 h1b y7d ff1 fs5 fc2 sc0 ls5 ws6a">capítulo com aplicações das equações de 1<span class="fs8 v5">a</span></div><div class="t m0 x26 h1a y7e ff1 fs8 fc2 sc0 ls3b">.<span class="fs5 ls5 ws6a v4">ordem, análise qualitativa das equações autônomas e existência e</span></div><div class="t m0 x18 h8 y7f ff1 fs5 fc2 sc0 ls5 ws0">unicidade de soluções.</div><div class="t m0 x1a h17 y80 ff1 fs5 fc2 sc0 ls5 ws6b">As equações lineares de 2<span class="fs8 v5">a</span></div><div class="t m0 x1d h1a y81 ff1 fs8 fc2 sc0 ls3c">.<span class="fs5 ls5 ws6b v4">ordem é o assunto do Capítulo 2.<span class="_1d blank"> </span>Aqui o estudo tanto das equações homogêneas</span></div><div class="t m0 x18 h8 y82 ff1 fs5 fc2 sc0 ls5 ws6c">como das equações não homogêneas é feito inicialmente no caso geral e depois no caso particular em que os</div><div class="t m0 x18 h8 y83 ff1 fs5 fc2 sc0 ls5 ws6d">coe\ufb01cientes são constantes.<span class="_22 blank"> </span>O capítulo contém também oscilações.<span class="_22 blank"> </span>O capítulo termina com soluções em série</div><div class="t m0 x18 h8 y84 ff1 fs5 fc2 sc0 ls5 ws6e">de potências em torno de <span class="ff7 ls3d">t</span><span class="fs8 ls3e v1">0</span><span class="ff8 fs9 ls3f">=</span>0 no caso em que este ponto é ordinário e mudanças de variáveis em equações</div><div class="t m0 x18 h1b y85 ff1 fs5 fc2 sc0 ls5 ws0">de 2<span class="fs8 v5">a</span></div><div class="t m0 x27 h1a y86 ff1 fs8 fc2 sc0 ls40">.<span class="fs5 ls5 ws65 v4">ordem.</span></div><div class="t m0 x1a h8 y87 ff1 fs5 fc2 sc0 ls5 ws6f">O Capítulo 3 trata da transformada de Laplace.<span class="_1b blank"> </span>O objetivo é r<span class="_1 blank"></span>esolver problemas de valor inicial para</div><div class="t m0 x18 h17 y88 ff1 fs5 fc2 sc0 ls5 ws70">equações lineares de 2<span class="fs8 v5">a</span></div><div class="t m0 x28 h1a y89 ff1 fs8 fc2 sc0 ls41">.<span class="fs5 ls5 ws70 v4">ordem tanto com o termo não homogêneo contínuo, quanto descontínuo.<span class="_1d blank"> </span>T<span class="_14 blank"></span>erminamos</span></div><div class="t m0 x18 h8 y32 ff1 fs5 fc2 sc0 ls5 ws0">o capítulo com a transformada de Laplace do delta de Dirac e com a convolução.</div><a class="l" data-dest-detail='[742,"XYZ",14.173,259.688,null]'><div class="d m1" style="border-style:none;position:absolute;left:304.401000px;bottom:280.597000px;width:6.974000px;height:8.926000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf9" class="pf w0 h0" data-page-no="9"><div class="pc pc9 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x18 y34 w4 h13" alt="" src="https://files.passeidireto.com/5a32494f-ee17-40d1-8874-e3139abd510a/bg9.png"><div class="t m0 x18 h14 y35 ff6 fs5 fc0 sc0 ls5 ws71">Apresentação <span class="fsb fc3">ix</span></div><div class="t m0 x1a h8 yd ff1 fs5 fc2 sc0 ls5 ws72">No Capítulo 4 o estudo de sistemas de equações diferenciais linear<span class="_1 blank"></span>es é feito usando diagonalização de ma-</div><div class="t m0 x18 h8 ye ff1 fs5 fc2 sc0 ls5 ws73">trizes.<span class="_8 blank"> </span>O caso 2<span class="_23 blank"> </span><span class="ff4 fs9 ls42">×</span>2 é tratado em separado com detalhe.<span class="_8 blank"> </span>O capítulo termina com os sistemas não homogêneos</div><div class="t m0 x18 h8 y66 ff1 fs5 fc2 sc0 ls5 ws0">e o uso da transformada de Laplace.</div><div class="t m0 x1a h8 y8a ff1 fs5 fc2 sc0 ls5 ws74">T<span class="_14 blank"></span>odos os exercícios estão resolvidos no \ufb01nal do capitulo corr<span class="_1 blank"></span>espondente.<span class="_d blank"> </span>Uma coisa que acho importante</div><div class="t m0 x18 h8 y8b ff1 fs5 fc2 sc0 ls5 ws75">é somente ler a solução de um exercício depois de ter tentado ver<span class="_1 blank"></span>dadeiramente resolvê-lo.<span class="_11 blank"> </span>É como quando</div><div class="t m0 x18 h8 y8c ff1 fs5 fc2 sc0 ls5 ws76">lhe dão um enigma para decifrar<span class="_2 blank"></span>.<span class="_8 blank"> </span>Se lhe contarem logo a solução, você a esquecerá logo depois.<span class="_8 blank"> </span>Quanto mais</div><div class="t m0 x18 h8 y8d ff1 fs5 fc2 sc0 ls5 ws77">tempo você \ufb01car tentando decifrar antes de lhe contarem a solução, tanto mais tempo você se lembrará da</div><div class="t m0 x18 h8 y8e ff1 fs5 fc2 sc0 ls5">solução.</div><div class="t m0 x1a h15 y8f ff1 fs5 fc2 sc0 ls5 ws78">Os desenhos e grá\ufb01cos foram feitos usando o M<span class="_24 blank"> </span><span class="fs7 ls43 ws79">A<span class="_2 blank"></span>TLAB<span class="ffc fsc ls44 v5">r<span class="ff4 fsa fc3 ls45">\u2217</span></span><span class="fs5 ls5 ws78">com o pacote GAAL e o Maxima também com</span></span></div><div class="t m0 x18 h8 y90 ff1 fs5 fc2 sc0 ls5 ws7a">o pacote GAAL disponíveis no site do autor (<span class="ff2 fc1 ws7b">http://www.mat.ufmg.br/\u02dcregi</span>).<span class="_1b blank"> </span>Neste site também estão</div><div class="t m0 x18 h8 y91 ff1 fs5 fc2 sc0 ls5 ws0">disponíveis páginas interativas para o estudo de oscilações, equações parciais, séries de Fourier e outr<span class="_1 blank"></span>os.</div><div class="t m0 x1a h8 y92 ff1 fs5 fc2 sc0 ls5 ws7c">Gostaria de agradecer ao professor Helder C. Rodrigues pelas fr<span class="_1 blank"></span>utíferas discussões, aos professores Rogé-</div><div class="t m0 x18 h8 y93 ff1 fs5 fc2 sc0 ls5 ws7d">rio S. Mol, Antônio Gaspar Ruas, Francisco Dutenhefner<span class="_2 blank"></span>, Grey Er<span class="_1 blank"></span>cole, Hamilton P<span class="_0 blank"></span>. Bueno, Antônio Zumpano,</div><div class="t m0 x18 h8 y94 ff1 fs5 fc2 sc0 ls5 ws7e">Marcelo T<span class="_14 blank"></span>. Cunha,<span class="_8 blank"> </span>Jorge Sabatucci, Regina Radich,<span class="_8 blank"> </span>Marcelo Mar<span class="_1 blank"></span>chesin,<span class="_8 blank"> </span>Ricardo T<span class="_14 blank"></span>akahashi, Lúcia Brasil,<span class="_8 blank"> </span>Ar-</div><div class="t m0 x18 h8 y95 ff1 fs5 fc2 sc0 ls5 ws7f">mando G. M. Neves e Carlos A. Arteaga pelas críticas e sugestões que possibilitaram o aperfeiçoamento do</div><div class="t m0 x18 h8 y96 ff1 fs5 fc2 sc0 ls5 ws0">presente texto.</div><div class="t m0 x1a h1c y97 ff4 fsd fc2 sc0 ls46">\u2217<span class="ff1 fs7 ls5 ws0 v7">MA<span class="_2 blank"></span>TLAB é marca registrada de The Mathworks, Inc.</span></div><div class="t m0 x18 h8 y54 ff6 fs5 fc0 sc0 ls5 ws0">Julho 2013<span class="_1e blank"> </span><span class="ff1 fc4 ws3d">GoBack GoForward<span class="_1f blank"> </span></span><span class="fc1">Reginaldo J. Santos</span></div><a class="l" data-dest-detail='[3,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:487.989000px;bottom:427.878000px;width:9.070000px;height:14.097000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[9,"XYZ",32.106,51.984,null]'><div class="d m1" style="border-style:none;position:absolute;left:279.924000px;bottom:276.350000px;width:6.624000px;height:13.880000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l"><div class="d m1" style="border-style:none;position:absolute;left:224.125000px;bottom:264.395000px;width:148.443000px;height:12.010000px;background-color:rgba(255,255,255,0.000001);"></div></a><div class="d m1" style="border-style:none;position:absolute;left:210.159000px;bottom:12.736000px;width:36.064000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div><div class="d m1" style="border-style:none;position:absolute;left:249.488000px;bottom:12.736000px;width:53.080000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div><a class="l"><div class="d m1" style="border-style:none;position:absolute;left:422.191000px;bottom:12.736000px;width:74.868000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pfa" class="pf w0 h0" data-page-no="a"><div class="pc pca w0 h0"><img fetchpriority="low" loading="lazy" class="bi x18 y34 w4 h13" alt="" src="https://files.passeidireto.com/5a32494f-ee17-40d1-8874-e3139abd510a/bga.png"><div class="t m0 x18 h14 y55 ff6 fsb fc3 sc0 ls47">x<span class="fs5 fc0 ls5">Apresentação</span></div><div class="t m0 x18 h1d y98 ff6 fse fc0 sc0 ls5 ws0">Sugestão de Cronograma par<span class="_1 blank"></span>a 60 Horas</div><div class="t m0 x26 h8 y99 ff1 fs5 fc2 sc0 ls5 ws0">Capítulo 1<span class="_25 blank"> </span>20 aulas</div><div class="t m0 x26 h8 y9a ff1 fs5 fc2 sc0 ls5 ws0">Capítulo 2<span class="_25 blank"> </span>20 aulas</div><div class="t m0 x26 h8 y9b ff1 fs5 fc2 sc0 ls5 ws0">Capítulo 3<span class="_25 blank"> </span>10 aulas</div><div class="t m0 x26 h8 y9c ff1 fs5 fc2 sc0 ls5 ws0">Capítulo 4<span class="_25 blank"> </span>10 aulas</div><div class="t m0 x26 h8 y9d ff1 fs5 fc2 sc0 ls5 ws0">T<span class="_14 blank"></span>otal<span class="_26 blank"> </span>60 aulas</div><div class="t m0 x21 h8 y54 ff6 fs5 fc0 sc0 ls5 ws0">Introdução às Equações Diferenciais Ordinárias<span class="_16 blank"> </span><span class="ff1 fc4 ws3d">GoBack GoForward<span class="_17 blank"> </span></span>Julho 2013</div><a class="l" data-dest-detail='[3,"XYZ",14.173,411.024,null]'><div class="d m1" style="border-style:none;position:absolute;left:13.177000px;bottom:428.027000px;width:6.894000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div></a><div class="d m1" style="border-style:none;position:absolute;left:210.159000px;bottom:12.736000px;width:36.064000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div><div class="d m1" style="border-style:none;position:absolute;left:249.488000px;bottom:12.736000px;width:53.080000px;height:13.948000px;background-color:rgba(255,255,255,0.000001);"></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
Compartilhar