<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/cc116f21-d16b-4582-b505-1f1637b5f0a6/bg1.png"><div class="c x0 y1 w2 h2"><div class="t m0 x1 h3 y2 ff1 fs0 fc0 sc0 ls11 ws0">EA<span class="_0 blank"></span>E0<span class="_0 blank"></span>207<span class="_0 blank"></span>:<span class="_1 blank"> </span>M<span class="_0 blank"></span>ate<span class="_0 blank"></span>mát<span class="_0 blank"></span>ica A<span class="_0 blank"></span>pli<span class="_0 blank"></span>cad<span class="_0 blank"></span>a à Eco<span class="_0 blank"></span>no<span class="_0 blank"></span>mia</div><div class="t m0 x2 h4 y3 ff1 fs1 fc0 sc0 ls11 ws1">Aula 19:<span class="_2 blank"> </span>Equações Diferenciais Sepa<span class="_0 blank"></span>ráveis e</div><div class="t m0 x3 h4 y4 ff1 fs1 fc0 sc0 ls11 ws2">Equações Diferencias de Segunda Ordem</div><div class="t m0 x4 h4 y5 ff1 fs1 fc1 sc0 ls11 ws3">Ma<span class="_0 blank"></span>rcos Y. Nakaguma</div><div class="t m0 x5 h4 y6 ff1 fs1 fc1 sc0 ls11 ws4">11/10/2017</div><div class="t m0 x6 h5 y7 ff1 fs2 fc1 sc0 ls11">1</div></div><div class="c x0 y8 w2 h2"><div class="t m0 x7 h6 y9 ff1 fs3 fc1 sc0 ls11 ws5">E<span class="_0 blank"></span>q<span class="_0 blank"></span>u<span class="_3 blank"></span>a<span class="_0 blank"></span>ç<span class="_0 blank"></span>õ<span class="_0 blank"></span>e<span class="_0 blank"></span>s S<span class="_0 blank"></span>e<span class="_0 blank"></span>p<span class="_3 blank"></span>a<span class="_3 blank"></span>r<span class="_0 blank"></span>á<span class="_0 blank"></span>v<span class="_0 blank"></span>e<span class="_0 blank"></span>is d<span class="_3 blank"></span>e P<span class="_3 blank"></span>ri<span class="_0 blank"></span>m<span class="_3 blank"></span>e<span class="_0 blank"></span>i<span class="_0 blank"></span>ra<span class="_2 blank"> </span>O<span class="_3 blank"></span>r<span class="_0 blank"></span>d<span class="_0 blank"></span>e<span class="_0 blank"></span>m</div><div class="t m0 x6 h5 ya ff1 fs2 fc1 sc0 ls11">2</div></div><div class="c x0 yb w2 h2"><div class="t m0 x8 h3 yc ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções S<span class="_0 blank"></span>epa<span class="_3 blank"></span>rá<span class="_0 blank"></span>veis d<span class="_0 blank"></span>e Pr<span class="_0 blank"></span>ime<span class="_0 blank"></span>ira<span class="_0 blank"></span>-Ord<span class="_0 blank"></span>em</div><div class="t m0 x9 h4 yd ff1 fs1 fc1 sc0 ls11 ws7">As <span class="fc2 ws8">equações diferenciais sepa<span class="_0 blank"></span>ráveis <span class="fc1 ws9">p ossuem<span class="_4 blank"> </span>a<span class="_4 blank"> </span>seguinte<span class="_5 blank"> </span>fo<span class="_0 blank"></span>rma:</span></span></div><div class="t m0 xa h7 ye ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 xa h8 yf ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ls1 v1">g</span><span class="ff3 fs4 ls2 v1">(</span><span class="ls3 v1">y</span><span class="ff3 fs4 ls4 v1">)</span><span class="ls5 v1">h</span><span class="ff3 fs4 ls6 v1">(</span><span class="ls7 v1">t</span><span class="ff3 fs4 ls8 v1">)</span><span class="ff4 v1">,</span></div><div class="t m0 x9 h4 y10 ff1 fs1 fc1 sc0 ls11 ws2">i.e.<span class="_2 blank"> </span>o lado direito da equação p<span class="_6 blank"> </span>o<span class="_6 blank"> </span>de ser escrito como o p<span class="_0 blank"></span>ro<span class="_6 blank"> </span>duto de um</div><div class="t m0 x9 h9 y11 ff1 fs1 fc1 sc0 ls11 wsb">fato<span class="_0 blank"></span>r que dep<span class="_6 blank"> </span>ende somente de <span class="ff2 ls9">y<span class="ff4 lsa">,</span><span class="fc2 lsb v0">g<span class="ff3 fs4 ls6">(</span><span class="ls3">y<span class="ff3 fs4 ls6">)</span><span class="ff4 fc1 lsc">,</span></span></span></span><span class="ws1 v0">e de um outro fator que</span></div><div class="t m0 x9 h9 y12 ff1 fs1 fc1 sc0 ls11 wsc">dep ende<span class="_5 blank"> </span>somente<span class="_4 blank"> </span>de<span class="_5 blank"> </span><span class="ff2 lsd">t<span class="ff4 lse">,</span><span class="fc2 lsf v0">h<span class="ff3 fs4 ls2">(</span><span class="ls7">t<span class="ff3 fs4 ls10">)</span></span></span></span><span class="ff4 v0">.</span></div><div class="t m0 x6 h5 y13 ff1 fs2 fc1 sc0 ls11">3</div></div></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 x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/cc116f21-d16b-4582-b505-1f1637b5f0a6/bg2.png"><div class="c x0 y1 w2 h2"><div class="t m0 x8 h3 y14 ff1 fs0 fc0 sc0 ls11 wse">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções S<span class="_0 blank"></span>epa<span class="_3 blank"></span>rá<span class="_0 blank"></span>veis d<span class="_0 blank"></span>e Pr<span class="_0 blank"></span>ime<span class="_0 blank"></span>ira<span class="_0 blank"></span>-Ord<span class="_0 blank"></span>em</div><div class="t m0 x9 h4 y15 ff1 fs1 fc2 sc0 ls11 ws2">Exemplo 1:<span class="_2 blank"> </span><span class="fc1 ws1">As seguintes equações diferenciais são sepa<span class="_0 blank"></span>ráveis:</span></div><div class="t m0 xa h7 y16 ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 x5 ha y17 ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ls12 v1">y</span><span class="ff1 fs2 ls13 v2">2</span><span class="ff5 ls14 v3">\ue000</span><span class="ls15 v1">t</span><span class="ff1 fs2 ls16 v2">2</span><span class="ff3 fs4 ls17 v1">+</span><span class="ls18 v1">t</span><span class="ff5 v3">\ue001</span></div><div class="t m0 xb h7 y18 ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 xb hb y19 ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ls19 v1">e</span><span class="fs2 ls1a v2">y</span><span class="ls19 v1">e</span><span class="fs2 v2">t</span></div><div class="t m0 xc h7 y1a ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 xc hc y1b ff2 fs1 fc1 sc0 ls11 wsf">dt <span class="ff3 fs4 ls1b v1">=<span class="ls6 v0">(</span></span><span class="ls1c v1">y<span class="ff3 fs4 ls17">+<span class="ff1 fs1 ls1d">1</span><span class="ls1e v0">)</span></span></span><span class="ff1 v4">1</span></div><div class="t m0 xd h7 y1b ff2 fs1 fc1 sc0 ls11">t</div><div class="t m0 xe h7 y1c ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 xe hd y1d ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ls9 v1">y</span><span class="ff1 fs2 ls1f v2">2</span><span class="ff3 fs4 ls20 v1">+</span><span class="ff1 v1">1</span></div><div class="t m0 x6 h5 y1e ff1 fs2 fc1 sc0 ls11">4</div></div><div class="c x0 y8 w2 h2"><div class="t m0 x8 h3 y1f ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções S<span class="_0 blank"></span>epa<span class="_3 blank"></span>rá<span class="_0 blank"></span>veis d<span class="_0 blank"></span>e Pr<span class="_0 blank"></span>ime<span class="_0 blank"></span>ira<span class="_0 blank"></span>-Ord<span class="_0 blank"></span>em</div><div class="t m0 x9 h4 y20 ff1 fs1 fc2 sc0 ls11 ws2">Exemplo 2:<span class="_2 blank"> </span><span class="fc1 ws1">As seguintes equações diferenciais são não-sepa<span class="_0 blank"></span>ráveis:</span></div><div class="t m0 xf h7 y21 ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 xf he y22 ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ls12 v1">y</span><span class="ff1 fs2 ls21 v2">2</span><span class="ff3 fs4 ls20 v1">+</span><span class="lsd v1">t</span><span class="ff1 fs2 v2">2</span></div><div class="t m0 x10 h7 y23 ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 x10 hf y24 ff2 fs1 fc1 sc0 ls11 wsf">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ls22 v1">a</span><span class="ff3 fs4 ls6 v1">(</span><span class="ls23 v1">t</span><span class="ff3 fs4 ls8 v1">)</span><span class="ls24 v1">y<span class="ff3 fs4 ls25">+</span><span class="ls26">b<span class="ff3 fs4 ls6 v0">(</span><span class="ls23">t</span></span></span><span class="ff3 fs4 v1">)</span></div><div class="t m0 xc h7 y25 ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 xc h10 y26 ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls27 v1">=</span><span class="ws10 v1">t<span class="_0 blank"></span>y <span class="ff3 fs4 ls20">+</span><span class="ls28">t<span class="ff1 fs2 ls29 v5">2</span><span class="ls12">y<span class="ff1 fs2 ls11 v5">2</span></span></span></span></div><div class="t m0 x6 h5 y27 ff1 fs2 fc1 sc0 ls11">5</div></div><div class="c x0 yb w2 h2"><div class="t m0 x8 h3 yc ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções S<span class="_0 blank"></span>epa<span class="_3 blank"></span>rá<span class="_0 blank"></span>veis d<span class="_0 blank"></span>e Pr<span class="_0 blank"></span>ime<span class="_0 blank"></span>ira<span class="_0 blank"></span>-Ord<span class="_0 blank"></span>em</div><div class="t m0 x9 h4 y28 ff1 fs1 fc1 sc0 ls11 ws1">Dada uma equação sepa<span class="_0 blank"></span>rável:</div><div class="t m0 xa h7 y29 ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 xa h8 y2a ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ls1 v1">g</span><span class="ff3 fs4 ls2 v1">(</span><span class="ls3 v1">y</span><span class="ff3 fs4 ls4 v1">)</span><span class="ls5 v1">h</span><span class="ff3 fs4 ls6 v1">(</span><span class="ls7 v1">t</span><span class="ff3 fs4 ls8 v1">)</span><span class="ff4 v1">,</span></div><div class="t m0 x9 h4 y2b ff1 fs1 fc1 sc0 ls11 ws1">re-escreva-a como:<span class="_7 blank"> </span><span class="v6">1</span></div><div class="t m0 x11 hf y2c ff2 fs1 fc1 sc0 ls2a">g<span class="ff3 fs4 ls6 v0">(</span><span class="ls3">y<span class="ff3 fs4 ls2b v0">)</span><span class="ls11 ws11 v1">dy <span class="ff3 fs4 ls27">=</span><span class="ls5">h<span class="ff3 fs4 ls6 v0">(</span><span class="ls23">t<span class="ff3 fs4 ls8 v0">)</span><span class="ls11">dt</span></span></span></span></span></div><div class="t m0 x9 h4 y2d ff1 fs1 fc1 sc0 ls11 ws12">Integrando o lado esquerdo em relação a <span class="ff2 ls2c">y</span><span class="ws13">e o lado direito em relação</span></div><div class="t m0 x9 h4 y2e ff1 fs1 fc1 sc0 ls2d">a<span class="ff2 ls2e">t</span><span class="ls11 ws2">, obtemos:<span class="_8 blank"> </span><span class="ff5 fs5 ls2f v7">Z</span><span class="v8">1</span></span></div><div class="t m0 x12 h11 y2f ff2 fs1 fc1 sc0 ls2a">g<span class="ff3 fs4 ls2 v0">(</span><span class="ls3">y<span class="ff3 fs4 ls30 v0">)</span><span class="ls11 ws14 v1">dy <span class="ff3 fs4 ls1b">=<span class="ff5 fs5 ls31 v9">Z</span></span></span><span class="ls5 v1">h</span><span class="ff3 fs4 ls6 v1">(</span><span class="ls23 v1">t</span><span class="ff3 fs4 ls8 v1">)</span><span class="ls11 ws15 v1">dt <span class="ff3 fs4 ls20">+</span>k</span></span></div><div class="t m0 x9 h12 y30 ff1 fs1 fc1 sc0 ls11 ws16">Resolva esta equação pa<span class="_0 blank"></span>ra <span class="ff2 ls32">y<span class="ff3 fs4 ls6 v0">(</span><span class="ls7">t<span class="ff3 fs4 ls8 v0">)</span></span></span><span class="ff4">.</span></div><div class="t m0 x6 h5 y31 ff1 fs2 fc1 sc0 ls11">6</div></div></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 x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/cc116f21-d16b-4582-b505-1f1637b5f0a6/bg3.png"><div class="c x0 y1 w2 h2"><div class="t m0 x8 h3 y14 ff1 fs0 fc0 sc0 ls11 wse">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções S<span class="_0 blank"></span>epa<span class="_3 blank"></span>rá<span class="_0 blank"></span>veis d<span class="_0 blank"></span>e Pr<span class="_0 blank"></span>ime<span class="_0 blank"></span>ira<span class="_0 blank"></span>-Ord<span class="_0 blank"></span>em</div><div class="t m0 x9 h4 y32 ff1 fs1 fc1 sc0 ls11 ws17">F<span class="_0 blank"></span>ormalmente, resolvemos uma equação sepa<span class="_0 blank"></span>rável da seguinte maneira:</div><div class="t m0 x9 h4 y33 ff1 fs1 fc1 sc0 ls11 ws1">Observe que p<span class="_6 blank"> </span>o<span class="_6 blank"> </span>demos re-exp<span class="_0 blank"></span>ressar a seguinte equação diferencial</div><div class="t m0 xa h7 y34 ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 xa hf y35 ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ls1 v1">g</span><span class="ff3 fs4 ls2 v1">(</span><span class="ls3 v1">y</span><span class="ff3 fs4 ls4 v1">)</span><span class="ls33 v1">h</span><span class="ff3 fs4 ls2 v1">(</span><span class="ls7 v1">t</span><span class="ff3 fs4 ls8 v1">)</span><span class="ff4 v1">,</span></div><div class="t m0 x9 h4 y36 ff1 fs1 fc1 sc0 ls11 ws18">como: <span class="va">1</span></div><div class="t m0 x5 h13 y37 ff2 fs1 fc1 sc0 ls2a">g<span class="ff3 fs4 ls2 v0">(</span><span class="ls3">y<span class="ff3 fs4 ls11 v0">)</span></span></div><div class="t m0 x13 h7 y38 ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 x13 h14 y37 ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ls33 v1">h</span><span class="ff3 fs4 ls6 v1">(</span><span class="ls7 v1">t</span><span class="ff3 fs4 v1">)</span></div><div class="t m0 x9 h4 y39 ff1 fs1 fc1 sc0 ls11 ws2">Integrando amb<span class="_6 blank"> </span>os os lados em relação a <span class="ff2 ls15">t</span>, obtemos:</div><div class="t m0 x14 h15 y3a ff5 fs5 fc1 sc0 ls34">Z<span class="ff1 fs1 ls11 vb">1</span></div><div class="t m0 x4 h13 y3b ff2 fs1 fc1 sc0 ls35">g<span class="ff3 fs4 ls10 v0">(</span><span class="ls3">y<span class="ff3 fs4 ls11 v0">)</span></span></div><div class="t m0 x15 h7 y3c ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 x15 h16 y3b ff2 fs1 fc1 sc0 ls11 ws19">dt <span class="ws1a v1">dt </span><span class="ff3 fs4 ls1b v1">=</span><span class="ff5 fs5 ls36 vc">Z</span><span class="ls37 v1">h</span><span class="ff3 fs4 ls6 v1">(</span><span class="ls23 v1">t</span><span class="ff3 fs4 ls8 v1">)</span><span class="ws15 v1">dt <span class="ff3 fs4 ls17">+</span><span class="ls38">k<span class="ff4 ls11">,</span></span></span></div><div class="t m0 x9 h17 y3d ff1 fs1 fc1 sc0 ls11 ws1b">onde, na notação de Leibniz,<span class="_2 blank"> </span><span class="ff2 fs2 ws1c vd">dy</span></div><div class="t m0 xb h18 y3e ff2 fs2 fc1 sc0 ls11 ws1d">dt<span class="_1 blank"> </span><span class="ff1 fs1 ws3 ve">é a derivada de <span class="ff2 ls39">y<span class="ff3 fs4 ls0 v0">=</span><span class="ls3a v0">y<span class="ff3 fs4 ls6 v0">(</span><span class="ls7">t<span class="ff3 fs4 ls4 v0">)</span><span class="ff4 ls11">.</span></span></span></span></span></div><div class="t m0 x6 h5 y3f ff1 fs2 fc1 sc0 ls11">7</div></div><div class="c x0 y8 w2 h2"><div class="t m0 x8 h3 y1f ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções S<span class="_0 blank"></span>epa<span class="_3 blank"></span>rá<span class="_0 blank"></span>veis d<span class="_0 blank"></span>e Pr<span class="_0 blank"></span>ime<span class="_0 blank"></span>ira<span class="_0 blank"></span>-Ord<span class="_0 blank"></span>em</div><div class="t m0 x9 h4 y40 ff1 fs1 fc1 sc0 ls11 ws2">P<span class="_0 blank"></span>elo méto<span class="_6 blank"> </span>do da substituição (mudança de variável), segue que dado</div><div class="t m0 x9 h9 y41 ff2 fs1 fc1 sc0 ls3b">y<span class="ff3 fs4 ls0 v0">=</span><span class="ls32 v0">y<span class="ff3 fs4 ls2 v0">(</span><span class="ls23">t<span class="ff3 fs4 ls6 v0">)</span><span class="ff1 ls11 ws1b">, então:</span></span></span></div><div class="t m0 x16 h19 y42 ff2 fs1 fc1 sc0 ls11 ws1e">dy <span class="ff3 fs4 ls3c v0">=</span><span class="v1">dy</span></div><div class="t m0 x17 h1a y43 ff2 fs1 fc1 sc0 ls11 ws1f">dt <span class="v1">dt</span></div><div class="t m0 x9 h4 y44 ff1 fs1 fc1 sc0 ls11 ws1">Assim, p<span class="_6 blank"> </span>o<span class="_6 blank"> </span>demos re-escrever a integral anterio<span class="_0 blank"></span>r como:</div><div class="t m0 x18 h15 y45 ff5 fs5 fc1 sc0 ls2f">Z<span class="ff1 fs1 ls11 vb">1</span></div><div class="t m0 x12 h1b y46 ff2 fs1 fc1 sc0 ls2a">g<span class="ff3 fs4 ls2 v0">(</span><span class="ls3">y<span class="ff3 fs4 ls30 v0">)</span><span class="ls11 ws14 v1">dy <span class="ff3 fs4 ls3d">=<span class="ff5 fs5 ls3e v9">Z</span></span></span><span class="ls3f v1">h</span><span class="ff3 fs4 ls6 v1">(</span><span class="ls23 v1">t</span><span class="ff3 fs4 ls8 v1">)</span><span class="ls11 ws15 v1">dt <span class="ff3 fs4 ls20">+</span>k</span></span></div><div class="t m0 x9 h1c y47 ff1 fs1 fc1 sc0 ls11 ws16">Resolva esta equação pa<span class="_0 blank"></span>ra <span class="ff2 ls32">y<span class="ff3 fs4 ls6 v0">(</span><span class="ls7">t<span class="ff3 fs4 ls8 v0">)</span></span></span><span class="ff4">.</span></div><div class="t m0 x6 h5 y48 ff1 fs2 fc1 sc0 ls11">8</div></div><div class="c x0 yb w2 h2"><div class="t m0 x8 h3 yc ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções S<span class="_0 blank"></span>epa<span class="_3 blank"></span>rá<span class="_0 blank"></span>veis d<span class="_0 blank"></span>e Pr<span class="_0 blank"></span>ime<span class="_0 blank"></span>ira<span class="_0 blank"></span>-Ord<span class="_0 blank"></span>em</div><div class="t m0 x9 h4 y49 ff1 fs1 fc2 sc0 ls11 ws2">Exemplo 1:<span class="_2 blank"> </span><span class="fc1">Considere a seguinte equação:</span></div><div class="t m0 xa h7 y4a ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 xa h1d y4b ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ff1 ls40 v1">3</span><span class="ff3 fs4 ls2 v1">(</span><span class="ff1 ls41 v1">2<span class="ff6 fs4 ls20">\ue000</span></span><span class="ls3 v1">y</span><span class="ff3 fs4 ls8 v1">)</span><span class="ls2e v1">t</span><span class="ff1 fs2 v2">2</span></div><div class="t m0 x9 h4 y4c ff1 fs1 fc1 sc0 ls11 ws16">Re-escreva esta exp<span class="_0 blank"></span>ressão como:</div><div class="t m0 xe h4 y4d ff1 fs1 fc1 sc0 ls11">1</div><div class="t m0 x11 h1e y4e ff3 fs4 fc1 sc0 ls10">(<span class="ff1 fs1 ls41 v0">2<span class="ff6 fs4 ls25">\ue000</span><span class="ff2 ls3">y</span></span><span class="ls30">)<span class="ff2 fs1 ls11 ws11 v1">dy </span><span class="ls0 v1">=<span class="ff1 fs1 ls11 ws4">3<span class="ff2 lsd">t</span><span class="fs2 ls42 v5">2</span><span class="ff2">dt</span></span></span></span></div><div class="t m0 x9 h4 y4f ff1 fs1 fc1 sc0 ls11 ws20">Integrando<span class="_5 blank"> </span>amb os<span class="_4 blank"> </span>os<span class="_5 blank"> </span>lados,<span class="_4 blank"> </span>obtemos:</div><div class="t m0 x19 h15 y50 ff5 fs5 fc1 sc0 ls43">Z<span class="ff1 fs1 ls11 vb">1</span></div><div class="t m0 x1a h1f y51 ff3 fs4 fc1 sc0 ls6">(<span class="ff1 fs1 ls41 v0">2<span class="ff6 fs4 ls20">\ue000</span><span class="ff2 ls3">y</span></span><span class="ls2b">)<span class="ff2 fs1 ls11 ws11 v1">dy </span><span class="ls44 v1">=</span><span class="ff5 fs5 ls31 vc">Z</span><span class="ff1 fs1 ls11 ws4 v1">3<span class="ff2 ls28">t<span class="ff1 fs2 ls42 v5">2</span><span class="ls11 ws21">dt <span class="ff4">,</span></span></span></span></span></div><div class="t m0 x9 h4 y52 ff1 fs1 fc1 sc0 ls11 ws2">o que implica que:</div><div class="t m0 x1b h20 y53 ff6 fs4 fc1 sc0 ls45">\ue000<span class="ff1 fs1 ls11 ws22">ln </span><span class="ls46">j<span class="ff1 fs1 ls41">2</span><span class="ls20">\ue000<span class="ff2 fs1 ls3">y</span><span class="ls47">j<span class="ff3 ls48">=<span class="ff2 fs1 ls15">t<span class="ff1 fs2 ls49 v5">3</span></span><span class="ls17">+<span class="ff2 fs1 ls11 ws4">k<span class="ff1 fs2 vf">1</span></span></span></span></span></span></span></div><div class="t m0 x6 h5 y54 ff1 fs2 fc1 sc0 ls11">9</div></div></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 x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/cc116f21-d16b-4582-b505-1f1637b5f0a6/bg4.png"><div class="c x0 y1 w2 h2"><div class="t m0 x8 h3 y14 ff1 fs0 fc0 sc0 ls11 wse">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções S<span class="_0 blank"></span>epa<span class="_3 blank"></span>rá<span class="_0 blank"></span>veis d<span class="_0 blank"></span>e Pr<span class="_0 blank"></span>ime<span class="_0 blank"></span>ira<span class="_0 blank"></span>-Ord<span class="_0 blank"></span>em</div><div class="t m0 x9 h4 y55 ff1 fs1 fc1 sc0 ls11 ws4">(Cont.)</div><div class="t m0 x1c h21 y56 ff7 fs6 fc0 sc0 ls4a">I<span class="ff1 fs7 fc1 ls11 ws23 vf">Assim, temos que a solução do p<span class="_0 blank"></span>roblema acima é dada p<span class="_6 blank"> </span>or:</span></div><div class="t m0 x1d h22 y57 ff6 fs8 fc1 sc0 ls4b">j<span class="ff1 fs7 ls4c">2</span><span class="ls4d">\ue000<span class="ff2 fs7 ls4e">y</span><span class="ls4f">j<span class="ff3 ls50">=<span class="ff2 fs7 ls51">e</span></span><span class="fs9 ls52 v10">\ue000<span class="ff2 fs2 ls53">t<span class="ff1 fsa ls54 v11">3</span></span><span class="ls55 v0">\ue000<span class="ff2 fs2 ls56">k<span class="ff1 fsa ls57 v12">1</span></span></span></span><span class="ls11 ws24 v0">$<span class="_1 blank"> </span>j<span class="ff1 fs7 ls4c">2</span><span class="ls4d">\ue000<span class="ff2 fs7 ls4e">y</span><span class="ls58">j<span class="ff3 ls59">=</span></span></span><span class="ff2 fs7 ws25">ke </span><span class="fs9 ls52 v10">\ue000<span class="ff2 fs2 ls5a">t<span class="ff1 fsa ls57 v11">3</span></span></span><span class="ls5b">$<span class="ff1 fs7 ls5c">2</span><span class="ls5d">\ue000<span class="ff2 fs7 ls5e">y</span><span class="ff3 ls5f">=</span><span class="ls60">\ue006</span></span></span><span class="ff2 fs7 ws26">k<span class="_0 blank"></span>e <span class="ff6 fs9 ls52 v10">\ue000</span><span class="fs2 ls61 v10">t</span><span class="ff1 fsa v13">3</span></span></span></span></span></div><div class="t m0 xc h23 y58 ff6 fs8 fc1 sc0 ls5b">)<span class="ff2 fs7 ls62">y</span><span class="ff3 ls63 v0">(</span><span class="ff2 fs7 ls64">t</span><span class="ff3 ls65 v0">)<span class="ls66 v0">=<span class="ff1 fs7 ls4c">2</span><span class="ls67">+<span class="ff2 fs7 ls11 ws25">k<span class="_0 blank"></span>e <span class="ff6 fs9 ls52 v10">\ue000</span><span class="fs2 ls68 v10">t</span><span class="ff1 fsa ls69 v13">3</span><span class="ff4">,</span></span></span></span></span></div><div class="t m0 x1e h24 y59 ff1 fs7 fc1 sc0 ls11 ws27">onde <span class="ff2 ls6a">k</span><span class="ws23">é uma constante a<span class="_0 blank"></span>rbitrária não-nula.</span></div><div class="t m0 x1f h5 y3f ff1 fs2 fc1 sc0 ls11 ws28">10</div></div><div class="c x0 y8 w2 h2"><div class="t m0 x8 h3 y1f ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções S<span class="_0 blank"></span>epa<span class="_3 blank"></span>rá<span class="_0 blank"></span>veis d<span class="_0 blank"></span>e Pr<span class="_0 blank"></span>ime<span class="_0 blank"></span>ira<span class="_0 blank"></span>-Ord<span class="_0 blank"></span>em</div><div class="t m0 x9 h4 y5a ff1 fs1 fc2 sc0 ls11 ws2">Exemplo 2:<span class="_2 blank"> </span><span class="fc1">Considere a seguinte equação:</span></div><div class="t m0 x20 h7 y5b ff2 fs1 fc1 sc0 ls11">dy</div><div class="t m0 x20 h25 y5c ff2 fs1 fc1 sc0 ls11 wsa">dt <span class="ff3 fs4 ls0 v1">=</span><span class="ls2e v1">t</span><span class="ff1 fs2 ls6b v2">2</span><span class="v1">y</span></div><div class="t m0 x9 h4 y5d ff1 fs1 fc1 sc0 ls11 ws16">Re-escreva esta exp<span class="_0 blank"></span>ressão como:</div><div class="t m0 x16 h4 y5e ff1 fs1 fc1 sc0 ls11">1</div><div class="t m0 x16 he y5f ff2 fs1 fc1 sc0 ls6c">y<span class="ls11 ws29 v1">dy </span><span class="ff3 fs4 ls0 v1">=</span><span class="ls2e v1">t</span><span class="ff1 fs2 ls42 v2">2</span><span class="ls11 v1">dt</span></div><div class="t m0 x9 h4 y60 ff1 fs1 fc1 sc0 ls11 ws2">Integrando amb<span class="_6 blank"> </span>os os lados, obtemos:</div><div class="t m0 x21 h15 y61 ff5 fs5 fc1 sc0 ls6d">Z<span class="ff1 fs1 ls11 vb">1</span></div><div class="t m0 xe h16 y62 ff2 fs1 fc1 sc0 ls6e">y<span class="ls11 ws2a v1">dy </span><span class="ff3 fs4 ls44 v1">=</span><span class="ff5 fs5 ls6f vc">Z</span><span class="ls28 v1">t</span><span class="ff1 fs2 ls42 v2">2</span><span class="ls11 ws21 v1">dt <span class="ff4">,</span></span></div><div class="t m0 x9 h4 y63 ff1 fs1 fc1 sc0 ls11 ws2">o que implica que:</div><div class="t m0 x22 h26 y64 ff1 fs1 fc1 sc0 ls11 ws2b">ln <span class="ff6 fs4 ls46 v0">j<span class="ff2 fs1 ls70">y</span><span class="ls71">j<span class="ff3 ls72">=</span></span></span><span class="ff2 v1">t</span></div><div class="t m0 x21 h4 y65 ff1 fs1 fc1 sc0 ls11">3</div><div class="t m0 x15 h5 y66 ff1 fs2 fc1 sc0 ls11">3</div><div class="t m0 x23 h27 y67 ff3 fs4 fc1 sc0 ls20">+<span class="ff2 fs1 ls11 ws4">k<span class="ff1 fs2 ls73 vf">1</span></span><span class="ff6 ls74">)<span class="ff2 fs1 ls3a">y</span></span><span class="ls6 v0">(</span><span class="ff2 fs1 ls23">t</span><span class="ls75 v0">)<span class="ls48 v0">=</span></span><span class="ff2 fs1 ls11 ws2c">k<span class="_0 blank"></span>e <span class="fsa v1">t</span></span></div><div class="t m0 x24 h28 y68 ff1 fsa fc1 sc0 ls11">3</div><div class="t m0 x25 h28 y69 ff1 fsa fc1 sc0 ls76">3<span class="ff4 fs1 ls11 va">,</span></div><div class="t m0 x9 h29 y6a ff1 fs1 fc1 sc0 ls11 ws2d">com <span class="ff2 ls77">k<span class="ff6 fs4 ls78 v0">2<span class="ff8 fs1 ls79">R<span class="ff4 ls11">.</span></span></span></span></div><div class="t m0 x1f h5 y6b ff1 fs2 fc1 sc0 ls11 ws28">11</div></div><div class="c x0 yb w2 h2"><div class="t m0 x26 h6 y6c ff1 fs3 fc1 sc0 ls11 ws5">E<span class="_3 blank"></span>q<span class="_0 blank"></span>u<span class="_0 blank"></span>a<span class="_0 blank"></span>ç<span class="_0 blank"></span>õ<span class="_0 blank"></span>e<span class="_0 blank"></span>s D<span class="_3 blank"></span>if<span class="_0 blank"></span>e<span class="_0 blank"></span>r<span class="_0 blank"></span>e<span class="_0 blank"></span>n<span class="_0 blank"></span>c<span class="_0 blank"></span>ia<span class="_3 blank"></span>is L<span class="_3 blank"></span>in<span class="_3 blank"></span>e<span class="_0 blank"></span>a<span class="_3 blank"></span>r<span class="_0 blank"></span>e<span class="_0 blank"></span>s d<span class="_0 blank"></span>e S<span class="_3 blank"></span>e<span class="_0 blank"></span>g<span class="_0 blank"></span>u<span class="_0 blank"></span>n<span class="_0 blank"></span>d<span class="_3 blank"></span>a O<span class="_3 blank"></span>rd<span class="_3 blank"></span>e<span class="_0 blank"></span>m</div><div class="t m0 x27 h2a y6d ff1 fs1 fc1 sc0 ls11 ws2e">(c<span class="ff6 fs4 ls7a">n</span><span class="ws20">co e\u2026cientes<span class="_4 blank"> </span>constantes)</span></div><div class="t m0 x1f h5 y13 ff1 fs2 fc1 sc0 ls11 ws28">12</div></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 x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/cc116f21-d16b-4582-b505-1f1637b5f0a6/bg5.png"><div class="c x0 y1 w2 h2"><div class="t m0 x8 h3 y14 ff1 fs0 fc0 sc0 ls11 ws2f">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções D<span class="_0 blank"></span>ife<span class="_0 blank"></span>ren<span class="_0 blank"></span>ciai<span class="_0 blank"></span>s Lin<span class="_0 blank"></span>ea<span class="_3 blank"></span>res d<span class="_0 blank"></span>e Seg<span class="_0 blank"></span>und<span class="_0 blank"></span>a-O<span class="_0 blank"></span>rde<span class="_0 blank"></span>m</div><div class="t m0 x9 h4 y6e ff1 fs1 fc1 sc0 ls11 ws16">As equações linea<span class="_0 blank"></span>res de segunda-ordem com coe\u2026cientes constantes</div><div class="t m0 x9 h4 y6f ff1 fs1 fc1 sc0 ls11 ws30">são<span class="_5 blank"> </span>dadas<span class="_4 blank"> </span>genericamente<span class="_5 blank"> </span>p or:</div><div class="t m0 x4 h2b y70 ff2 fs1 fc1 sc0 ls7b">a<span class="ls7c v1">d</span><span class="ff1 fs2 ls42 v14">2</span><span class="ls11 v1">y</span></div><div class="t m0 x12 h2c y71 ff2 fs1 fc1 sc0 ls11 ws31">dt <span class="ff1 fs2 ls7d v15">2</span><span class="ff3 fs4 ls20 v1">+</span><span class="ls7e v1">b</span><span class="v4">dy</span></div><div class="t m0 x28 h8 y71 ff2 fs1 fc1 sc0 ls11 ws32">dt <span class="ff3 fs4 ls20 v1">+</span><span class="ws33 v1">cy <span class="ff3 fs4 ls0">=</span><span class="ls2a">g<span class="ff3 fs4 ls6 v0">(</span><span class="ls7">t</span></span></span><span class="ff3 fs4 v1">)</span></div><div class="t m0 x9 h4 y72 ff1 fs1 fc1 sc0 ls11 ws34">Neste curso, discutiremos os seguintes casos:</div><div class="t m0 x29 h24 y73 ff2 fs7 fc0 sc0 ls7f">i<span class="ff4 ls80">.<span class="ff1 fc1 ls11 ws35">Equações <span class="fc2 ws36">homogêneas </span><span class="ws37">de segunda-o<span class="_0 blank"></span>rdem:</span></span></span></div><div class="t m0 x2a h2d y74 ff2 fs7 fc1 sc0 ls81">a<span class="ls82 v13">d</span><span class="ff1 fs2 ls6b v16">2</span><span class="ls11 v13">y</span></div><div class="t m0 xf h2e y75 ff2 fs7 fc1 sc0 ls11 ws38">dt <span class="ff1 fs2 ls83 v11">2</span><span class="ff3 fs8 ls84 v13">+</span><span class="ls85 v13">b</span><span class="ws39 v9">dy</span></div><div class="t m0 x2b h2f y76 ff2 fs7 fc1 sc0 ls11 ws3a">dt <span class="ff3 fs8 ls4d v13">+</span><span class="ws3b v13">cy <span class="ff3 fs8 ls86">=</span></span><span class="ff1 fc2 v13">0</span></div><div class="t m0 x2c h24 y77 ff2 fs7 fc0 sc0 ls11 ws3c">ii <span class="ff4 ls87">.</span><span class="ff1 fc1 ws35">Equações <span class="fc2 ws3d">não-homogêneas </span><span class="ws37">de segunda-o<span class="_0 blank"></span>rdem:</span></span></div><div class="t m0 x1a h2d y78 ff2 fs7 fc1 sc0 ls88">a<span class="ls89 v13">d</span><span class="ff1 fs2 ls42 v16">2</span><span class="ls11 v13">y</span></div><div class="t m0 x2a h30 y79 ff2 fs7 fc1 sc0 ls11 ws3e">dt <span class="ff1 fs2 ls8a v11">2</span><span class="ff3 fs8 ls67 v13">+</span><span class="ls85 v13">b</span><span class="v9">dy</span></div><div class="t m0 x2d h31 y7a ff2 fs7 fc1 sc0 ls11 ws3f">dt <span class="ff3 fs8 ls8b v13">+</span><span class="ws40 v13">cy <span class="ff3 fs8 ls8c">=</span></span><span class="fc2 ls8d v13">g<span class="ff3 fs8 ls8e">(</span><span class="ls8f">t<span class="ff3 fs8 ls11">)</span></span></span></div><div class="t m0 x1f h5 y7b ff1 fs2 fc1 sc0 ls11 ws28">13</div></div><div class="c x0 y8 w2 h2"><div class="t m0 x8 h3 y1f ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções D<span class="_0 blank"></span>ife<span class="_0 blank"></span>ren<span class="_0 blank"></span>ciai<span class="_0 blank"></span>s Ho<span class="_0 blank"></span>mo<span class="_0 blank"></span>gên<span class="_0 blank"></span>eas d<span class="_0 blank"></span>e Segu<span class="_0 blank"></span>nd<span class="_0 blank"></span>a-O<span class="_0 blank"></span>rde<span class="_0 blank"></span>m</div><div class="t m0 x9 h4 y7c ff1 fs1 fc1 sc0 ls11 ws41">Considere a seguinte equação <span class="fc2 ws42">homogênea </span><span class="ws3">de segunda-o<span class="_0 blank"></span>rdem:</span></div><div class="t m0 x12 h32 y7d ff2 fs1 fc1 sc0 ls7b">a<span class="ls7c v1">d</span><span class="ff1 fs2 ls42 v14">2</span><span class="ls11 v1">y</span></div><div class="t m0 x1a h33 y7e ff2 fs1 fc1 sc0 ls11 ws21">dt <span class="ff1 fs2 ls90 v15">2</span><span class="ff3 fs4 ls20 v1">+</span><span class="ls7e v1">b</span><span class="v4">dy</span></div><div class="t m0 x2e h34 y7e ff2 fs1 fc1 sc0 ls11 ws32">dt <span class="ff3 fs4 ls20 v1">+</span><span class="ws43 v1">cy <span class="ff3 fs4 ls0">=<span class="ff1 fs1 fc2 ls91">0<span class="fc1 ls11">(1)</span></span></span></span></div><div class="t m0 x9 h13 y7f ff1 fs1 fc1 sc0 ls11 ws1b">Nosso objetivo é encontra<span class="_0 blank"></span>r a <span class="fc2 ws44">solução geral </span><span class="ff2 ls32">y<span class="ff3 fs4 ls6 v0">(</span><span class="ls23">t<span class="ff3 fs4 ls92 v0">)</span></span></span><span class="ws45">deste problema.</span></div><div class="t m0 x9 h4 y80 ff1 fs1 fc1 sc0 ls11 ws3">P<span class="_0 blank"></span>ara determina<span class="_0 blank"></span>rmos a solução geral de uma equação diferencial linear</div><div class="t m0 x9 h4 y81 ff1 fs1 fc1 sc0 ls11 ws2">de segunda-o<span class="_0 blank"></span>rdem, utilizaremos o seguinte teo<span class="_0 blank"></span>rema:</div><div class="t m0 x9 h4 y82 ff1 fs1 fc2 sc0 ls11 ws46">T<span class="_3 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws1">A solução geral de uma equação diferencial homogênea</span></div><div class="t m0 x9 h35 y83 ff1 fs1 fc1 sc0 ls11 ws47">linea<span class="_0 blank"></span>r de 2<span class="ff2 fs2 ls93 ve">a</span><span class="ws1">ordem é dada pela combinação linear de <span class="fc2 v0">quaisquer duas</span></span></div><div class="t m0 x9 h4 y84 ff1 fs1 fc2 sc0 ls11 ws48">soluções<span class="_5 blank"> </span>particula<span class="_0 blank"></span>res<span class="_4 blank"> </span>independentes<span class="fc1 v0">.</span></div><div class="t m0 x1f h5 ya ff1 fs2 fc1 sc0 ls11 ws28">14</div></div><div class="c x0 yb w2 h2"><div class="t m0 x8 h3 yc ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções D<span class="_0 blank"></span>ife<span class="_0 blank"></span>ren<span class="_0 blank"></span>ciai<span class="_0 blank"></span>s Ho<span class="_0 blank"></span>mo<span class="_0 blank"></span>gên<span class="_0 blank"></span>eas d<span class="_0 blank"></span>e Segu<span class="_0 blank"></span>nd<span class="_0 blank"></span>a-O<span class="_0 blank"></span>rde<span class="_0 blank"></span>m</div><div class="t m0 x9 h4 y85 ff1 fs1 fc1 sc0 ls11 ws16">V<span class="_0 blank"></span>amos de\u2026nir a equação:</div><div class="t m0 x11 h36 y86 ff2 fs1 fc1 sc0 ls94">a<span class="ff9 ls95">\u03bb<span class="ff1 fs2 ls96 v5">2</span><span class="ff3 fs4 ls25 v0">+</span></span><span class="ls97 v0">b<span class="ff9 ls98">\u03bb<span class="ff3 fs4 ls20">+</span></span><span class="ls99">c<span class="ff3 fs4 ls0">=</span><span class="ff1 ls11 ws4">0<span class="ff4 ls9a">,</span>(2)</span></span></span></div><div class="t m0 x9 h37 y87 ff1 fs1 fc1 sc0 ls11 ws49">denominada <span class="fc2 ws4a v0">equação ca<span class="_0 blank"></span>racterística <span class="fc1 ws4b">de <span class="ff3 fs4 ls2 v0">(</span><span class="ls9b">1<span class="ff3 fs4 ls8 v0">)</span><span class="ff4 ls9c">.</span></span><span class="ws20">Denote<span class="_4 blank"> </span>p o<span class="_0 blank"></span>r<span class="_4 blank"> </span><span class="ff9 fc2 ls95">\u03bb<span class="ff1 fs2 ls9d vf">1</span></span><span class="ls9e">e<span class="ff9 fc2 ls95">\u03bb<span class="ff1 fs2 ls9f vf">2</span></span></span>as</span></span></span></div><div class="t m0 x9 h4 y88 ff1 fs1 fc1 sc0 ls11 ws4c">raízes desta equação.</div><div class="t m0 x1f h5 y89 ff1 fs2 fc1 sc0 ls11 ws28">15</div></div></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 x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/cc116f21-d16b-4582-b505-1f1637b5f0a6/bg6.png"><div class="c x0 y1 w2 h2"><div class="t m0 x8 h3 y14 ff1 fs0 fc0 sc0 ls11 wse">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções D<span class="_0 blank"></span>ife<span class="_0 blank"></span>ren<span class="_0 blank"></span>ciai<span class="_0 blank"></span>s Ho<span class="_0 blank"></span>mo<span class="_0 blank"></span>gên<span class="_0 blank"></span>eas d<span class="_0 blank"></span>e Segu<span class="_0 blank"></span>nd<span class="_0 blank"></span>a-O<span class="_0 blank"></span>rde<span class="_0 blank"></span>m</div><div class="t m0 x8 h38 y8a ff1 fsb fc0 sc0 ls11 ws4d">Ra íz e s<span class="_5 blank"> </span>Re a is</div><div class="t m0 x9 h4 y8b ff1 fs1 fc2 sc0 ls11 ws1">T<span class="_3 blank"></span>eo<span class="_0 blank"></span>rema 1:<span class="_2 blank"> </span><span class="fc1 ws2">Considere a equação:</span></div><div class="t m0 x12 h39 y8c ff2 fs1 fc1 sc0 ls7b">a<span class="ls7c v1">d</span><span class="ff1 fs2 ls42 v14">2</span><span class="ls11 v1">y</span></div><div class="t m0 x1a h3a y8d ff2 fs1 fc1 sc0 ls11 ws21">dt <span class="ff1 fs2 ls90 v15">2</span><span class="ff3 fs4 ls20 v1">+</span><span class="ls7e v1">b</span><span class="v4">dy</span></div><div class="t m0 x2e h8 y8d ff2 fs1 fc1 sc0 ls11 ws32">dt <span class="ff3 fs4 ls20 v1">+</span><span class="ws33 v1">cy <span class="ff3 fs4 ls27">=</span><span class="ff1">0</span></span></div><div class="t m0 x9 h4 y8e ff1 fs1 fc1 sc0 ls11 ws45">e sup<span class="_6 blank"> </span>onha que as raízes <span class="ff9 ls95">\u03bb</span><span class="fs2 lsa0 vf">1</span><span class="ls9e">e<span class="ff9 ls95">\u03bb</span><span class="fs2 lsa1 vf">2</span></span><span class="ws1">da equação característica</span></div><div class="t m0 x9 h3b y8f ff2 fs1 fc1 sc0 ls94">a<span class="ff9 lsa2">\u03bb<span class="ff1 fs2 lsa3 v5">2</span><span class="ff3 fs4 ls20 v0">+</span></span><span class="lsa4 v0">b<span class="ff9 lsa5">\u03bb<span class="ff3 fs4 ls25">+</span></span><span class="ls99">c<span class="ff3 fs4 ls0">=</span><span class="ff1 ls11 ws2">0 sejam <span class="fc2 ws4">reais</span>.</span></span></span></div><div class="t m0 x29 h3c y90 ff2 fs7 fc0 sc0 ls7f">i<span class="ff4 ls80">.<span class="ff1 fc1 ls11 ws4e">Se <span class="ff9 lsa6">\u03bb</span><span class="fs2 lsa7 vf">1</span><span class="ff6 fs8 lsa8">6<span class="ff3 lsa9">=</span></span><span class="ff9 lsaa">\u03bb</span><span class="fs2 ls42 vf">2</span><span class="ws4f">, então a solução geral da equação acima é dada p<span class="_6 blank"> </span>o<span class="_0 blank"></span>r:</span></span></span></div><div class="t m0 xa h3d y91 ff2 fs7 fc1 sc0 lsab">y<span class="ff3 fs8 lsac v0">(</span><span class="ls8f">t<span class="ff3 fs8 ls65 v0">)<span class="ls59 v0">=</span></span><span class="ls11 ws39">k<span class="ff1 fs2 ls6b vf">1</span><span class="lsad">e<span class="ff9 fsc lsae ve">\u03bb</span><span class="ff1 fsa lsaf v17">1</span><span class="fs2 lsb0 ve">t</span><span class="ff3 fs8 ls4d v0">+</span></span><span class="v0">k<span class="ff1 fs2 ls42 vf">2</span><span class="lsad">e<span class="ff9 fsc lsb1 ve">\u03bb</span><span class="ff1 fsa lsaf v17">2</span></span><span class="fs2 ve">t</span></span></span></span></div><div class="t m0 x2c h3c y92 ff2 fs7 fc0 sc0 ls11 ws3c">ii <span class="ff4 ls87">.</span><span class="ff1 fc1 ws4e">Se <span class="ff9 lsa6">\u03bb</span><span class="fs2 lsa7 vf">1</span><span class="ff3 fs8 ls5f">=</span><span class="ff9 lsb2">\u03bb</span><span class="fs2 ls42 vf">2</span><span class="ws37">, então a solução geral da equação acima é dada p<span class="_6 blank"> </span>o<span class="_0 blank"></span>r:</span></span></div><div class="t m0 x11 h3e y93 ff2 fs7 fc1 sc0 lsb3">y<span class="ff3 fs8 ls8e v0">(</span><span class="ls8f">t<span class="ff3 fs8 lsb4 v0">)<span class="ls59 v0">=</span></span><span class="ls11 ws39">k<span class="ff1 fs2 ls42 vf">1</span><span class="lsb5">e<span class="ff9 fsc lsb6 ve">\u03bb</span><span class="ff1 fsa lsaf v17">1</span><span class="fs2 lsb7 ve">t</span><span class="ff3 fs8 ls5d v0">+</span></span><span class="v0">k<span class="ff1 fs2 lsb8 vf">2</span></span><span class="fc2 lsb9">t</span><span class="lsb5">e<span class="ff9 fsc lsba ve">\u03bb</span><span class="ff1 fsa lsaf v17">1</span></span><span class="fs2 ve">t</span></span></span></div><div class="t m0 x1f h5 y7b ff1 fs2 fc1 sc0 ls11 ws28">16</div></div><div class="c x0 y8 w2 h2"><div class="t m0 x8 h3 y1f ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções D<span class="_0 blank"></span>ife<span class="_0 blank"></span>ren<span class="_0 blank"></span>ciai<span class="_0 blank"></span>s Ho<span class="_0 blank"></span>mo<span class="_0 blank"></span>gên<span class="_0 blank"></span>eas d<span class="_0 blank"></span>e Segu<span class="_0 blank"></span>nd<span class="_0 blank"></span>a-O<span class="_0 blank"></span>rde<span class="_0 blank"></span>m</div><div class="t m0 x8 h38 y94 ff1 fsb fc0 sc0 ls11 ws50">Ca so<span class="_5 blank"> </span>1:<span class="_9 blank"> </span>Ra íz e s<span class="_5 blank"> </span>Rea i s<span class="_5 blank"> </span>Dis ti n ta s</div><div class="t m0 x9 h4 y95 ff1 fs1 fc2 sc0 ls11 ws2">Exemplo 1:<span class="_2 blank"> </span><span class="fc1 ws16">Resolva o seguinte p<span class="_0 blank"></span>roblema de valor inicial:</span></div><div class="t m0 x2f h3f y96 ff2 fs1 fc1 sc0 ls7c">d<span class="ff1 fs2 ls42 ve">2</span><span class="ls11">y</span></div><div class="t m0 x10 h40 y97 ff2 fs1 fc1 sc0 ls11 ws21">dt <span class="ff1 fs2 lsbb v15">2</span><span class="ff6 fs4 lsbc v1">\ue000</span><span class="v4">dy</span></div><div class="t m0 x30 h8 y97 ff2 fs1 fc1 sc0 ls11 ws32">dt <span class="ff6 fs4 ls20 v1">\ue000</span><span class="ff1 ws4 v1">2</span><span class="lsbd v1">y<span class="ff3 fs4 ls0">=</span></span><span class="ff1 ws4 v1">0<span class="ff4">,</span></span></div><div class="t m0 x9 h41 y98 ff1 fs1 fc1 sc0 ls11 ws51">onde <span class="ff2 lsbe">y<span class="ff3 fs4 ls10 v0">(</span></span><span class="lsbf v0">0<span class="ff3 fs4 lsc0 v0">)<span class="lsc1 v0">=</span></span><span class="ls11 ws2">3 e<span class="_2 blank"> </span><span class="ff2 fs2 ws28 vd">dy</span></span></span></div><div class="t m0 x31 h42 y99 ff2 fs2 fc1 sc0 ls11 ws1d">dt<span class="_2 blank"> </span><span class="ff3 fs4 ls10 ve">(</span><span class="ff1 fs1 lsbf ve">0</span><span class="ff3 fs4 lsc0 ve">)<span class="lsc1 v0">=</span></span><span class="ff1 fs1 ws4 ve">0<span class="ff4">.</span></span></div><div class="t m0 x1c h43 y9a ff7 fs6 fc0 sc0 ls4a">I<span class="ff1 fs7 fc1 ls11 ws4f vf">Primeiro, devemos encontra<span class="_0 blank"></span>r a solução geral do problema.<span class="_9 blank"> </span>A <span class="fc2 ws39">equação</span></span></div><div class="t m0 x1e h44 y9b ff1 fs7 fc2 sc0 ls11 ws52">ca<span class="_0 blank"></span>racterística <span class="fc1 ws53 v0">asso<span class="_6 blank"> </span>ciada à equação diferencial acima é dada p<span class="_6 blank"> </span>or:</span></div><div class="t m0 x20 h45 y9c ff9 fs7 fc1 sc0 lsa6">\u03bb<span class="ff1 fs2 lsc2 v10">2</span><span class="ff6 fs8 lsc3">\ue000</span><span class="lsc4">\u03bb<span class="ff6 fs8 ls4d">\ue000</span><span class="ff1 lsc5">2<span class="ff3 fs8 ls66">=</span><span class="ls11">0</span></span></span></div><div class="t m0 x1e h3c y9d ff1 fs7 fc1 sc0 ls11 ws53">As raízes desta equação são <span class="ff9 lsc6">\u03bb</span><span class="fs2 lsa7 vf">1</span><span class="ff3 fs8 ls66">=</span><span class="ws37">2 e<span class="_5 blank"> </span><span class="ff9 lsa6">\u03bb</span><span class="fs2 lsa7 vf">2</span><span class="ff3 fs8 lsc7">=<span class="ff6 lsc8">\ue000</span></span><span class="ws39">1<span class="ff4 lsc9">.</span></span>Po<span class="_0 blank"></span>rtanto, a <span class="fc2 ws39 v0">solução</span></span></div><div class="t m0 x1e h24 y9e ff1 fs7 fc2 sc0 ls11 ws54">geral <span class="fc1">é:</span></div><div class="t m0 x2a h46 y9f ff2 fs7 fc1 sc0 lsab">y<span class="ff3 fs8 lsca v0">(</span><span class="ls8f">t<span class="ff3 fs8 ls65 v0">)<span class="ls66 v0">=</span></span><span class="ls11 ws39">k<span class="ff1 fs2 ls42 vf">1</span><span class="lscb">e<span class="ff1 fs2 lscc v10">2<span class="ff2 lscd">t</span></span><span class="ff3 fs8 lsce">+</span></span>k<span class="ff1 fs2 ls42 vf">2</span><span class="lsb5">e<span class="ff6 fs9 lscf v10">\ue000</span></span><span class="fs2 v10">t</span></span></span></div><div class="t m0 x1f h5 ya0 ff1 fs2 fc1 sc0 ls11 ws28">17</div></div><div class="c x0 yb w2 h2"><div class="t m0 x8 h3 yc ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções D<span class="_0 blank"></span>ife<span class="_0 blank"></span>ren<span class="_0 blank"></span>ciai<span class="_0 blank"></span>s Ho<span class="_0 blank"></span>mo<span class="_0 blank"></span>gên<span class="_0 blank"></span>eas d<span class="_0 blank"></span>e Segu<span class="_0 blank"></span>nd<span class="_0 blank"></span>a-O<span class="_0 blank"></span>rde<span class="_0 blank"></span>m</div><div class="t m0 x8 h38 ya1 ff1 fsb fc0 sc0 ls11 ws50">Ca so<span class="_5 blank"> </span>1:<span class="_9 blank"> </span>Ra íz e s<span class="_5 blank"> </span>Rea i s<span class="_5 blank"> </span>Dis ti n ta s</div><div class="t m0 x9 h4 ya2 ff1 fs1 fc1 sc0 ls11 ws4">(Cont.)</div><div class="t m0 x1c h47 ya3 ff7 fs6 fc0 sc0 ls4a">I<span class="ff1 fs7 fc1 ls11 ws55 vf">P<span class="_0 blank"></span>ara encontra<span class="_0 blank"></span>r a <span class="fc2 ws37">solução particula<span class="_0 blank"></span>r<span class="fc1 ws53">, devemos avaliar as funções:</span></span></span></div><div class="t m0 x2a h48 ya4 ff2 fs7 fc1 sc0 lsab">y<span class="ff3 fs8 lsca v0">(</span><span class="ls8f">t<span class="ff3 fs8 ls65 v0">)<span class="ls66 v0">=</span></span><span class="ls11 ws39">k<span class="ff1 fs2 ls42 vf">1</span><span class="lscb">e<span class="ff1 fs2 lscc v10">2<span class="ff2 lscd">t</span></span><span class="ff3 fs8 lsce">+</span></span>k<span class="ff1 fs2 ls42 vf">2</span><span class="lsb5">e<span class="ff6 fs9 lscf v10">\ue000</span></span><span class="fs2 v10">t</span></span></span></div><div class="t m0 x1e h24 ya5 ff1 fs7 fc1 sc0 lsd0">e<span class="ff2 ls11 v6">dy</span></div><div class="t m0 x21 h49 ya6 ff2 fs7 fc1 sc0 ls11 ws56">dt <span class="ff3 fs8 ls63 v13">(</span><span class="lsd1 v13">t</span><span class="ff3 fs8 lsb4 v13">)<span class="ls59 v0">=</span></span><span class="ff1 ws39 v13">2<span class="ff2">k<span class="ff1 fs2 lsd2 vf">1</span><span class="lsd3">e<span class="ff1 fs2 lsd4 v10">2<span class="ff2 lsd5">t</span></span><span class="ff6 fs8 ls84">\ue000</span></span>k<span class="ff1 fs2 ls42 vf">2</span><span class="lsad">e<span class="ff6 fs9 lsd6 v10">\ue000</span></span><span class="fs2 v10">t</span></span></span></div><div class="t m0 x1e h24 ya7 ff1 fs7 fc1 sc0 ls11 ws37">no p<span class="_6 blank"> </span>onto 0<span class="ff4 lsc9">.</span><span class="ws4f">Assim, obtemos as seguintes equações:</span></div><div class="t m0 x32 h3c ya8 ff2 fs7 fc1 sc0 lsb3">y<span class="ff3 fs8 ls8e v0">(</span><span class="ff1 lsd7">0<span class="ff3 fs8 lsb4 v0">)<span class="ls59 v0">=</span></span></span><span class="ls11 ws39">k<span class="ff1 fs2 lsd8 vf">1</span><span class="ff3 fs8 ls8b">+</span>k<span class="ff1 fs2 lsd9 vf">2</span><span class="ff3 fs8 ls59">=</span><span class="ff1">3</span></span></div><div class="t m0 x1e h24 ya9 ff1 fs7 fc1 sc0 lsda">e<span class="ff2 ls11 v6">dy</span></div><div class="t m0 x2a h4a yaa ff2 fs7 fc1 sc0 ls11 ws56">dt <span class="ff3 fs8 ls63 v13">(</span><span class="ff1 lsdb v13">0</span><span class="ff3 fs8 ls65 v13">)<span class="ls66 v0">=</span></span><span class="ff1 ws39 v13">2<span class="ff2">k<span class="ff1 fs2 lsc2 vf">1</span><span class="ff6 fs8 lsce">\ue000</span>k<span class="ff1 fs2 lsa7 vf">2</span><span class="ff3 fs8 ls59">=</span><span class="ff1">0</span></span></span></div><div class="t m0 x1c h4b yab ff7 fs6 fc0 sc0 ls4a">I<span class="ff1 fs7 fc1 ls11 ws37 vf">Resolvendo o sistema acima par<span class="_0 blank"></span>a <span class="ff2 ws39">k<span class="ff1 fs2 lsdc vf">1</span><span class="ff1 lsdd">e</span>k<span class="ff1 fs2 ls42 vf">2</span></span>, obtemos:<span class="_2 blank"> </span><span class="ff2 ws39">k</span><span class="fs2 lsa7 vf">1</span><span class="ff3 fs8 ls66">=</span>1 e <span class="ff2 ws39">k</span><span class="fs2 lsa7 vf">2</span><span class="ff3 fs8 ls66">=</span><span class="ws39">2<span class="ff4">.</span></span></span></div><div class="t m0 x1e h24 yac ff1 fs7 fc1 sc0 ls11 ws37">P<span class="_0 blank"></span>ortanto, a solução do p<span class="_0 blank"></span>roblema de valor inicial é:</div><div class="t m0 xe h4c yad ff2 fs7 fc1 sc0 lsb3">y<span class="ff3 fs8 ls63 v0">(</span><span class="lsde">t<span class="ff3 fs8 lsb4 v0">)<span class="ls59 v0">=</span></span><span class="lscb">e<span class="ff1 fs2 lsd4 v10">2<span class="ff2 lsdf">t</span></span><span class="ff3 fs8 ls8b">+</span><span class="ff1 ls11 ws39">2</span><span class="lsad">e<span class="ff6 fs9 lscf v10">\ue000</span><span class="fs2 ls11 v10">t</span></span></span></span></div><div class="t m0 x1f h5 yae ff1 fs2 fc1 sc0 ls11 ws28">18</div></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 x0 yaf w1 h4d" alt="" src="https://files.passeidireto.com/cc116f21-d16b-4582-b505-1f1637b5f0a6/bg7.png"><div class="c x0 y1 w2 h2"><div class="t m0 x8 h3 y14 ff1 fs0 fc0 sc0 ls11 wse">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções D<span class="_0 blank"></span>ife<span class="_0 blank"></span>ren<span class="_0 blank"></span>ciai<span class="_0 blank"></span>s Ho<span class="_0 blank"></span>mo<span class="_0 blank"></span>gên<span class="_0 blank"></span>eas d<span class="_0 blank"></span>e Segu<span class="_0 blank"></span>nd<span class="_0 blank"></span>a-O<span class="_0 blank"></span>rde<span class="_0 blank"></span>m</div><div class="t m0 x8 h38 y8a ff1 fsb fc0 sc0 ls11 ws57">Ca so<span class="_5 blank"> </span>2:<span class="_9 blank"> </span>Ra íz e s<span class="_5 blank"> </span>Rea i s<span class="_5 blank"> </span>Igua i s</div><div class="t m0 x9 h4 yb0 ff1 fs1 fc2 sc0 ls11 ws2">Exemplo 2:<span class="_2 blank"> </span><span class="fc1 ws45">Resolva o seguinte p<span class="_0 blank"></span>roblema de valor inicial:</span></div><div class="t m0 x1b h3f yb1 ff2 fs1 fc1 sc0 lse0">d<span class="ff1 fs2 ls42 ve">2</span><span class="ls11">y</span></div><div class="t m0 x33 h4e yb2 ff2 fs1 fc1 sc0 ls11 ws31">dt <span class="ff1 fs2 lse1 v15">2</span><span class="ff6 fs4 ls17 v1">\ue000</span><span class="ff1 lse2 v1">4</span><span class="v4">dy</span></div><div class="t m0 x34 h4f yb2 ff2 fs1 fc1 sc0 ls11 ws58">dt <span class="ff3 fs4 ls20 v1">+</span><span class="ff1 ws4 v1">4</span><span class="lse3 v1">y<span class="ff3 fs4 ls0">=</span></span><span class="ff1 ws4 v1">0<span class="ff4">,</span></span></div><div class="t m0 x9 h50 yb3 ff1 fs1 fc1 sc0 ls11 ws59">onde <span class="ff2 lse4">y<span class="ff3 fs4 ls10 v0">(</span></span><span class="lsbf v0">0<span class="ff3 fs4 lsc0 v0">)<span class="lsc1 v0">=</span></span><span class="ls11 ws2">2 e<span class="_2 blank"> </span><span class="ff2 fs2 ws1d vd">dy</span></span></span></div><div class="t m0 x31 h51 yb4 ff2 fs2 fc1 sc0 ls11 ws1d">dt<span class="_2 blank"> </span><span class="ff3 fs4 ls10 ve">(</span><span class="ff1 fs1 lsbf ve">0</span><span class="ff3 fs4 lsc0 ve">)<span class="lsc1 v0">=</span></span><span class="ff1 fs1 ws4 ve">5<span class="ff4">.</span></span></div><div class="t m0 x1c h47 yb5 ff7 fs6 fc0 sc0 ls4a">I<span class="ff1 fs7 fc1 ls11 ws53 vf">Primeiro, devemos encontra<span class="_0 blank"></span>r a solução geral do problema.<span class="_9 blank"> </span>A <span class="fc2 ws39">equação</span></span></div><div class="t m0 x1e h24 yb6 ff1 fs7 fc2 sc0 ls11 ws52">ca<span class="_0 blank"></span>racterística <span class="fc1 ws4f v0">asso<span class="_6 blank"> </span>ciada à equação diferencial acima é dada p<span class="_6 blank"> </span>or:</span></div><div class="t m0 xb h52 yb7 ff9 fs7 fc1 sc0 lsaa">\u03bb<span class="ff1 fs2 lse5 v10">2</span><span class="ff6 fs8 ls4d">\ue000</span><span class="ff1 lsdb">4</span><span class="lsc4">\u03bb<span class="ff3 fs8 ls67">+</span><span class="ff1 lse6">4<span class="ff3 fs8 ls66">=</span><span class="ls11">0</span></span></span></div><div class="t m0 x1e h3c yb8 ff1 fs7 fc1 sc0 ls11 ws5a">As raízes desta equação são <span class="ff9 lsc6">\u03bb</span><span class="fs2 lsa7 vf">1</span><span class="ff3 fs8 lse7">=</span><span class="ff9 lsc6">\u03bb</span><span class="fs2 lsd9 vf">2</span><span class="ff3 fs8 ls66">=</span><span class="ws39">2<span class="ff4 lse8">.</span><span class="ws37">Po<span class="_0 blank"></span>rtanto, a <span class="fc2 v0">solução geral <span class="fc1">é:</span></span></span></span></div><div class="t m0 x2a h53 yb9 ff2 fs7 fc1 sc0 lsb3">y<span class="ff3 fs8 ls63 v0">(</span><span class="ls8f">t<span class="ff3 fs8 ls65 v0">)<span class="ls66 v0">=</span></span><span class="ls11 ws39">k<span class="ff1 fs2 ls29 vf">1</span><span class="lse9">e<span class="ff1 fs2 lsd4 v10">2<span class="ff2 lsdf">t</span></span><span class="ff3 fs8 ls8b">+</span></span>k<span class="ff1 fs2 ls42 vf">2</span><span class="ws5b">te <span class="ff1 fs2 lscc v10">2</span><span class="fs2 v10">t</span></span></span></span></div><div class="t m0 x1f h5 yba ff1 fs2 fc1 sc0 ls11 ws28">19</div></div><div class="c x0 y8 w2 h2"><div class="t m0 x8 h3 y1f ff1 fs0 fc0 sc0 ls11 ws6">Eq<span class="_0 blank"></span>ua<span class="_0 blank"></span>ções D<span class="_0 blank"></span>ife<span class="_0 blank"></span>ren<span class="_0 blank"></span>ciai<span class="_0 blank"></span>s Ho<span class="_0 blank"></span>mo<span class="_0 blank"></span>gên<span class="_0 blank"></span>eas d<span class="_0 blank"></span>e Segu<span class="_0 blank"></span>nd<span class="_0 blank"></span>a-O<span class="_0 blank"></span>rde<span class="_0 blank"></span>m</div><div class="t m0 x8 h38 y94 ff1 fsb fc0 sc0 ls11 ws5c">Ca s o<span class="_5 blank"> </span>2:<span class="_9 blank"> </span>Raíz e s<span class="_5 blank"> </span>Re a is<span class="_5 blank"> </span>Igu a is</div><div class="t m0 x9 h4 ybb ff1 fs1 fc1 sc0 ls11 ws4">(Cont.)</div><div class="t m0 x1c h47 ybc ff7 fs6 fc0 sc0 ls4a">I<span class="ff1 fs7 fc1 ls11 ws55 vf">P<span class="_0 blank"></span>ara encontra<span class="_0 blank"></span>r a <span class="fc2 ws37">solução particula<span class="_0 blank"></span>r<span class="fc1 ws53">, devemos avaliar as funções:</span></span></span></div><div class="t m0 x2a h48 ybd ff2 fs7 fc1 sc0 lsb3">y<span class="ff3 fs8 ls63 v0">(</span><span class="ls8f">t<span class="ff3 fs8 ls65 v0">)<span class="ls66 v0">=</span></span><span class="ls11 ws39">k<span class="ff1 fs2 ls42 vf">1</span><span class="lsd3">e<span class="ff1 fs2 lsd4 v10">2<span class="ff2 lsdf">t</span></span><span class="ff3 fs8 ls8b">+</span></span>k<span class="ff1 fs2 ls42 vf">2</span><span class="ws5b">te <span class="ff1 fs2 lscc v10">2</span><span class="fs2 v10">t</span></span></span></span></div><div class="t m0 x1e h24 ybe ff1 fs7 fc1 sc0 lsea">e<span class="ff2 ls11 v6">dy</span></div><div class="t m0 x35 h49 ybf ff2 fs7 fc1 sc0 ls11 ws5d">dt <span class="ff3 fs8 ls8e v13">(</span><span class="ls8f v13">t</span><span class="ff3 fs8 ls65 v13">)<span class="ls66 v0">=</span></span><span class="ff1 ws39 v13">2<span class="ff2">k<span class="ff1 fs2 ls6b vf">1</span><span class="lsd3">e<span class="ff1 fs2 lsd4 v10">2<span class="ff2 lsb7">t</span></span><span class="ff3 fs8 ls5d">+</span></span>k<span class="ff1 fs2 ls6b vf">2</span><span class="lseb">e<span class="ff1 fs2 lscc v10">2<span class="ff2 lsb0">t</span></span><span class="ff3 fs8 ls4d">+</span></span></span>2<span class="ff2">k</span><span class="fs2 ls6b vf">2</span><span class="ff2 ws5e">te </span><span class="fs2 lscc v10">2</span></span><span class="fs2 v18">t</span></div><div class="t m0 x1e h24 yc0 ff1 fs7 fc1 sc0 ls11 ws37">no p<span class="_6 blank"> </span>onto 0.<span class="_9 blank"> </span>Assim, obtemos as seguintes equações:</div><div class="t m0 x20 h3c yc1 ff2 fs7 fc1 sc0 ls62">y<span class="ff3 fs8 ls8e v0">(</span><span class="ff1 lsd7">0<span class="ff3 fs8 lsec v0">)<span class="ls50 v0">=</span></span></span><span class="ls11 ws39">k<span class="ff1 fs2 lsa7 vf">1</span><span class="ff3 fs8 ls66">=</span><span class="ff1">2</span></span></div><div class="t m0 x1e h24 yc2 ff1 fs7 fc1 sc0 lsda">e<span class="ff2 ls11 v6">dy</span></div><div class="t m0 x2a h4a yc3 ff2 fs7 fc1 sc0 ls11 ws56">dt <span class="ff3 fs8 ls63 v13">(</span><span class="ff1 lsdb v13">0</span><span class="ff3 fs8 ls65 v13">)<span class="ls66 v0">=</span></span><span class="ff1 ws39 v13">2<span class="ff2">k<span class="ff1 fs2 lsc2 vf">1</span><span class="ff3 fs8 lsce">+</span>k<span class="ff1 fs2 lsa7 vf">2</span><span class="ff3 fs8 ls59">=</span><span class="ff1">5</span></span></span></div><div class="t m0 x1c h54 yc4 ff7 fs6 fc0 sc0 ls4a">I<span class="ff1 fs7 fc1 ls11 ws37 vf">Resolvendo o sistema acima par<span class="_0 blank"></span>a <span class="ff2 ws39">k<span class="ff1 fs2 lsdc vf">1</span><span class="ff1 lsdd">e</span>k<span class="ff1 fs2 ls42 vf">2</span></span>, obtemos:<span class="_2 blank"> </span><span class="ff2 ws39">k</span><span class="fs2 lsa7 vf">1</span><span class="ff3 fs8 ls66">=</span>2 e <span class="ff2 ws39">k</span><span class="fs2 lsa7 vf">2</span><span class="ff3 fs8 ls66">=</span><span class="ws39">1<span class="ff4">.</span></span></span></div><div class="t m0 x1e h24 yc5 ff1 fs7 fc1 sc0 ls11 ws37">P<span class="_0 blank"></span>ortanto, a solução do p<span class="_0 blank"></span>roblema de valor inicial é:</div><div class="t m0 xf h4c yc6 ff2 fs7 fc1 sc0 lsab">y<span class="ff3 fs8 lsac v0">(</span><span class="ls8f">t<span class="ff3 fs8 ls65 v0">)<span class="ls59 v0">=</span></span><span class="ff1 ls11 ws39">2</span><span class="lscb">e<span class="ff1 fs2 lscc v10">2<span class="ff2 lscd">t</span></span><span class="ff3 fs8 ls67">+</span><span class="ls11 ws5b">te <span class="ff1 fs2 lsd4 v10">2</span><span class="fs2 v10">t</span></span></span></span></div><div class="t m0 x1f h5 yc7 ff1 fs2 fc1 sc0 ls11 ws28">20</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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