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O objetivo desse material é auxiliá-los para uma melhor compreensão dos métodos vistos em sala de aula da disciplina de equações diferenciais ordinárias. Método das variáveis separáveis- Lista de exercícios resolvida 1. 13 x dx dy dxxdy 13 dxxdy 13 Cxxy 2 3 2 yxxC 2 3 2 yxxC 2 3 2 2. 0 dyxdxy dyxdxy y dy x dx y dy x dx Cyx ||ln||ln ||ln||ln yxC y xC ln y x C ee ln y xC 3. 04 dy y xdxx y dydx x x 4 y dydx x x 4 Cyxx ln4 3 248 3 yxxC ln4 3 248 3 yxxC ln3424243 3 yxxC ln342424 3 4. 0sectansectan dyxydxyx dyxydxyx sectansectan dy y ydx x x sec tan sec tan dyydxx sinsin dyydxx sinsin Cyx coscos xyC coscos 5. 011 222 dyxdxyx dyxdxyx 222 11 22 2 1 1 y dydx x x 22 2 1 1 y dydx x x Cy x x arcsin1 y x xC arcsin1 6. 01 dxydyx dxydyx 1 1 x dx y dy 1x dx y dy Cxy |1|lnln |1|lnln xyC 1 ln x yC 1 ln x y C ee 1 x yC 1 xCy 7. 2 2 1 1 x y dx dy 22 11 x dx y dy 22 11 x dx y dy Cxy arctanarctan xyC arctanarctan 8. x dx dy 5sin dxxdy 5sin dxxdy 5sin Cxy 5cos 5 1 9. 03 dyedx x dyedx x3 dy e dx x 3 dye dx x3 Cye x 3 3 1 Cey x 3 3 1 Cey x 3 3 1 10. 61 x dx dyx dxxdyx 61 dx x xdy 1 6 dx x xdy 1 6 Cxxy 1ln5 11. yyx 4' y dx dyx 4 x dx y dy 4 x dx y dy 4 Cxy lnln 4 1 xyC lnln 4 1 x yC 4 1 ln x y C ee 4 1 ln x yC 4 1 4 1 yCx 4Cxy 4Cxy 12. 2 3 x y dx dy 23 x dx y dy 23 x dx y dy C xy 1 2 1 2 C xy 2212 Cxy 12 2 13. x yx dy dx 1 22 dyy x dxx 2 2 1 Cyx x 3 ln1 3 x x yC ln1 3 3 x x yC ln333 3 x x yC ln333 xxxyCx ln333 Cxxyxx 3ln33 14. yxe dx dy 23 yx ee dx dy 23 dxe e dy x y 3 2 dxee dy x y 3 2 Cee xy 32 3 1 2 1 Cee xy 623 32 Cee xy 32 23 dyy x dxx 2 2 1 15. 024 22 dxxyxdyyxy dxxyxdyyxy 22 24 dxyxdyxy 22 24 22 42 x dxx y dyy 22 42 x dxx y dyy Cxy 22 4ln 2 12ln 2 1 Cxy 24ln2ln 22 22 4ln2ln xyC 2 2 4 2ln x yC 2 2 4 2ln x y C ee 2 2 4 2 x yC 22 24 yxC 16. dxxdyxy 12 1 2 x dxxdyy 12 x dxxdyy Cxxy 1ln2 17. 21ln x y dy dxxy y dyydxxx 2 2 1ln y dyydxxx 2 2 1ln Cyyyxxx ln2 29 ln 3 233 18. 00,cos1sin1 ycomdyxdxxe y dyxdxxe y cos1sin1 1cos1 sin ye dy x dxx 11cos1 sin ye dy x dxx y y e e dy x dxx 1cos1 sin y y e dye x dxx 1cos1 sin y y e dye x dxx 1cos1 sin Cex y 1lncos1ln xeC y cos1ln1ln xeC yee cos11ln xeC y cos11 0cos11 0 eC 4C xe y cos114 19. 10,412 ycomdyydxxy 1 4 2 y dyydxx 1 4 2y dyydxx Cyx 12 22 12 22 yxC 1102 22 C 2C 122 22 yx 221 22 xy 20. 14,14 2 xcomxdy dx dyxdx 14 2 dyx dx 14 2 dyx dx 14 2 Cyx arctan 4 1 yxC arctan 4 1 4 1arctan 4 1 C 444 1 C 16 3 C yx arctan 4 1 16 3 yx 4arctan 4 3 4 34arctan yx 4 34tan)tan(arctan yx 4 34tan yx 21. 11,'2 ycomxyyyx xyy dx dyx 2 dxxyydyx 2 dxxydyx 12 dx x x y dy 2 1 dx x x y dy 2 1 Cx x y ln1ln C x xy 1lnln C x xy 1ln x xyC 1ln 1 111ln C 1C 11ln x xy 11ln xxy ee 11 xexy 22. 2y dx dyee xx dxydyee xx 2 xx ee dx y dy 2 xx ee dx y dy 2 Ce y x arctan1 23. 2pp dt dp dtppdp 2 dtpp dp 2 dtpp dp 2 Ct p p 1ln Ct p p 1 ln Ctp p ee 1 ln Ct ee p p 1 tCe p p 1 24. xyyx dx dy 1 xyx dx dy 111 yx dx dy 11 dxyxdy 11 dxxy dy 1 1 dxxy dy 1 1 Cxxy 2 1ln 2 25. 20,22 ycomyxxy dx dy 121 yyx dx dy 21 xy dx dy dxxy dy 2 1 dxxy dy 2 1 Cxxy 2 2 1ln 2 xxyC 2 2 1ln 2 02 2 012ln 2 C 3lnC 3ln2 2 1ln 2 xxy xxy 2 2 3ln1ln 2 xxy 2 23 1ln 2 xxy ee 2 23 1ln 2 x x ey 2 2 2 3 1 xx ey 2 2 2 31 13 2 2 2 xx ey 26. 4 0,0sin1cos ycomdyyedxy x dyyedxy x sin1cos dyy y e dx x cos sin 1 dyy y e dx x cos sin 1 Cye x cosln1ln yeC x cosln1ln y eC x cos 1ln 4cos 1ln 0 eC 2 2 2lnC 22lnC y e x cos 1ln22ln y e x ee cos 1ln22ln y e x cos 122 ye x sec122 27. 00, 1 10 2 ycom xdx dy dx x dy 1 10 2 dx x dy 1 10 2 Cxy arctan10 xyC arctan10 0arctan100 C 0C 0arctan10 xy xy arctan10 Equações Diferenciais Homogêneas- Lista de exercícios resolvida. 1. 0222 dyxydxyx 0222 xdvvdxvxxdxvxx 021 22 xdvvdxvdxvx 021 2 xdvvdxvdxv 0221 22 xvdvdxvdxv 0231 2 xvdvdxv xvdvdxv 231 2 231 2 v vdv x dx 231 2 v vdv x dx Cvx 231ln 3 1ln x yv C x yx 2 231ln 3 1ln C x yx 331lnln3 2 2 C x yx 331lnln 2 2 3 C x yxx 2 22 3 3ln Cx yxx ee 2 22 3 3ln C x yxx 2 22 3 3 Cyxx 22 3 Cxyx 23 3 02 2222 xdvvdxvxdxvxx 2. 042 dyyxdxyx 042 vdxxdvvxxdxvxx 0412 vdxxdvvdxvx 0442 2 dxvxvdvvdxxdvdxv 0442 2 dxvxvdvvdxxdvdxv 041422 2 dvvxdxvv dvvxdxvv 41422 2 dvvv v x dx 2422 41 dv vv v x dx 2422 41 Cvvx 2422ln 2 1ln x yv C x y x yx 2 2422ln 2 1ln C x y x yx 2422lnln2 2 2 C x y x yx 2422lnln 2 2 2 C x yxyxx 2 22 2 422ln Cyxyx ee 22 422ln Cyxyx 22 422 3. 022 dyxydxyx 0222 vdxxdvvxxdxxvx 01 22 vdxxdvvdxvx 01 2 vdxxdvvdxv 01 22 dxvxvdvdxv 0 xvdvdx xvdvdx vdv x dx vdvx dx Cvx 2 ln 2 C x yx 2 2 2 ln C x yx 2ln2 2 2 C x y x ee 2 2 2ln Cx y eex 2 2 2 2 2 2 x y Cex 4. 21,023 22 xeycomdyxydxyx 023 222 vdxxdvxvxdxvxx 0231 22 vdxxdvvdxvx 0231 2 vdxxdvvdxv 02231 22 dxvxvdvdxv 021 2 xvdvdxv xvdvdxv 21 2 21 2 v vdv x dx 21 2 v vdv x dx Cvx 21lnln C x yx 2 21lnln C x y x 2 21 ln x yv x yv C 2 22 11 2ln 3 8lnC 3 8ln 1 ln 2 2 x y x 3 8ln1 ln 2 2 ee x y x 3 8 1 2 2 x y x 2 218 3 x yx x x y 8 31 2 2 x x y 8 31 5. 0 dyxdxyx 0 vdxxdvxdxxvx 01 vdxxdvdxvx 01 vdxxdvdxv 01 vdxxdvdxv 0 xdvdx xdvdx dv x dx dvx dx Cvx ln x yv C x yx ln Cxyxx ln Cxyxx ln 6. 02 dyxydxx 02 vdxxdvxxvdxx 02 vdxxdvvdxx 02 vdxxdvvdx 022 2 dxvvdxxdvvxdvdx 0221 2 dvxvxdxvv dvvxdxvv 221 2 221 2 vv dvv x dx 221 2 vv dvv x dx C v vvx 1 121ln 2 1ln 2 x yv C x yx y x yx 1 121ln 2 1ln 2 2 C x xyx yxyxx 222lnln2 2 22 C xy x x yxyxx 222lnln 2 22 2 Cyx xyxyx 222ln 22 C yx xyx 22ln 2 yx yxCxyx 22ln2 yxCxyxyx 22ln2 C xy x x yxyxx 222ln 2 22 2 yxCxyxyx ln 7. 022 dyxdxyxy 0222 vdxxdvxdxxxvvx 022 vdxxdvdxvvx 02 vdxxdvdxvv 02 vdxxdvdxvv 02 xdvdxv xdvdxv 2 2v dv x dx 2v dv x dx C v x 1ln x yv C y xx ln y Cyxx ln Cyxxy ln Cyxxy ln 8. xy xy dx dy dxxydyxy dxxxvdyxxv dxvxvdxxdvvx 11 dxvvdxxdvv 11 012 dxvdxvvdxxdvxvdv 0112 dvvxdxv dvvxdxv 112 1 1 2 v dvv x dx 1 1 2v dvv x dx Cvvx arctan1ln 2 1ln 2 x yv C x y x yx 2arctan21lnln2 2 2 C x y x xyx 2arctan2lnln 2 22 2 C x y x xyx arctan2ln 2 22 2 C x yxy arctan2ln 22 9. 0 dyxyxdxy 0 vdxxdvxvxxxvdx 0 vdxxdvvxxxvdx 01 vdxxdvvvdxx 01 vdxxdvvvdx 0 vdxxdvdvvxdxvvvdx 01 dvvxdxvv dvvxdxvv 1 vv dvv x dx 1 vv dvv x dx 1 C v vx 2lnln x yv C x yx yx 2lnln C x yx yx 2lnln C x yx yx 2ln 2 2 2 2ln C x y y 22 4ln C x yy 22 4ln C y xy yCxyy 22 4ln yCyyx 22ln4 22ln4 Cyyx 10. dyyxdxyx 332 32 vdxxdvvxxdxxvx 3332 32 vdxxdvvxdxvx 333 32 vdxxdvvdxv 332 0332 34 vdxdvxvdxvxdvdxv dvvxdxvv 34 3 vv dvv x dx 4 33 vv dvv x dx 4 33 Cvvvvx 1ln 12 11ln 4 11ln 4 1ln 34 x yv C x y x y x y x yx 1ln 12 11ln 4 11ln 4 1ln 3 3 4 4 11. y x x y dx dy xy xy dx dy 22 dxxyxydy 22 dxxvxvdxxdvxvx 222 dxvxvdxxdvvx 1222 dxvvdxxdvv 12 0122 dxvdxvxvdv 01 dxxvdv x dxvdv x dxvdv Cxv ln 2 2 x yv Cx x y ln 2 2 2 Cx x y 2ln22 2 Cx x y ln2 2 12. y x yex dy dxy 2 4 dyyexydx y x 2 4 vdxxdvxvexxvdx v 2 4 vdxxdvvexxvdx v 2 41 vdxxdvvevdx v 2 41 044 222 vdxxdvdxevdvxvevdx vv dvvexdxev vv 222 414 v v ev dvve x dx 22 2 4 41 v v ev dvve x dx 22 2 4 41 Cevx v 2 8 1lnln x yv Ce x yx y x 2 8 1lnln Ce x yx y x 2 8 1lnln Ce x yx y x 8ln8 2 y x eCy 2 ln8 y x eCy 2 ln8 13. 0cot xdydx x yxy 0cot vdxxdvxdx x xvxxv 0cot vdxxdvdxvvx 0cot vdxxdvdxvv 0cot vdxxdvdxvv xdvdxv cot v dv x dx cot v dv x dx cot Cvx seclnln x yv C x yx seclnln C x yx seclnln C x y x sec ln Cx y x ee sec ln C x y x sec C x yx cos 14. 022 xydydxyxyx 0222 vdxxdvxvxdxvxxvxx 01 22 vdxxdvvdxvvx 01 2 vdxxdvvdxvv 01 22 dxvxvdvdxvv xvdvdxv 1 v vdv x dx 1 v vdv x dx 1 Cvvx 1lnlnx yv C x y x yx 1lnln C x y x yxx lnln C x y x yx x ln C x y yx x 2 ln C x y yx x ee 2 ln Cx y ee yx x 2 x y Ce yx x 2 x y Ceyxx 2 x y Ce yxx 2 x y eCxyx 2 15. 21,332 yxy dx dyxy dxxydyxy 332 dxxvxvdxxdvvxx 33322 dxvxvdxxdvvx 13323 dxvvdxxdvv 132 01332 dxvdxvdvxv dvxvdx 2 dvv x dx 2 dvvx dx 2 Cvx 3 ln 3 x yv C x yx 3 3 3 ln C x yx 3 3 3 ln C 3 3 13 21ln 3 8 C 3 8 3 ln 3 3 x yx 3 8 3 ln3 3 33 x yxx 8 ln3 3 33 x yxx 333 8ln3 xyxx 16. 21,32 22 yyxy dx dyx dxyxydyx 22 32 dxvxxvxvdxxdvx 222 32 dxvvxvdxxdvx 222 32 dxvvvdxxdv 232 0322 2 dxvvvdxxdv xdvdxvv 22 vv dv x dx 2 2 vv dv x dx 2 2 C v vx 1 ln2ln x yv C xy yx 2 lnln x xy yC lnln 2 C x xy y 2 2 ln C xyx y 2 2 ln C 2 2 121 2ln 4lnC 4lnln 2 2 xyx y 4ln ln 2 2 ee xyx y 42 2 xyx y 22 4 xyxy 24 xyxy xyxy 2 xxxyy 2 2 3 2 1 2 xyxy 17. 01,0 ydyxedxyex x y x y dyxedxyex x y x y vdxxdvxedxxvex x xv x xv vdxxdvxedxvex vv 1 vdxxdvedxve vv 1 01 dxvedvxedxve vvv dvxedx v dve x dx v dvex dx v Cex v ln x yv Cex x y ln Cex x y ln Ce 1 0 1ln 1C 1ln x y ex 1ln x y ex 18. 11,43 22 ydyxyxdxxyy vdxxdvxvxxdxxvxvx 222 43 vdxxdvvxdxvvx 43 222 vdxxdvvdxvv 432 0443 22 vdxxdvdxvxvdvdxvv dvvxdxv 4 v dvv x dx 4 v dvv x dx 4 Cvvx ln4ln x yv C x y x yx ln4ln C x y x yx ln4ln C 1 1 1 1ln41ln 1C 1ln4ln x y x yx xy x yxxx ln4ln 0ln4ln xy x yxxx 19. 11,2321 yyxyx dx dyxyx yxyx dx dyxyx 2 3 2 1 dxyxyxdyxyx 2 3 2 1 dxxvxvxxvdxxdvxvxx 2 3 2 3 2 1 dxxvxxvvdxxdvvxx 2 3 dxvvxvdxxdvvx 11 2 3 dxvvvdxxdvv 11 2 3 012 3 dxvvvdxxdvdxvvdvvx dvvxdx 1 dvv x dx 1 dvvx dx 1 Cvvx 3 2ln 2 3 x yv C x y x yx 2 3 2 3 3 2ln C x y x yx 2 3 2 3 3 2ln C 2 3 2 3 13 12 1 11ln 3 5 C 3 5 3 2ln 2 3 2 3 x y x yx 3 5 3 23ln3 2 3 2 3 2 1 2 3 x yyxxx 5 23ln3 2 3 2 3 2 1 2 3 x yyxxx 2 3 2 3 2 1 2 3 523ln3 xyyxxx 20. 10,0222 ydyyxyxdxy 022222 vdxxdvvxxvxxdxvx 01 222 vdxxdvvvdxvx 01 22 vdxxdvvvdxv 03222 dxvdvxvdxvxvdvvdxxdvdxv dvvvxdxvvv 223 12 vvv dvvv x dx 23 2 2 1 vvv dvvv x dx 23 2 2 1 C v vx 1 1lnln x yv C x yx yx 1 1lnln C yx x x yx lnln C yx x x yx ln C yx xy ln C 10 01ln 0C 0ln yx xy 0 ln yx xyyx 0ln xyyx 21. 1 2 1,2 yy dx dyxyyx ydxdyxyyx 2 xvdxvdxxdvxvxvxx 22 xvdxvdxxdvvxvxx 222 xvdxvdxxdvvvxx 2 xvdxvdxxdvvvx 21 vdxvdxxdvvv 21 022 vdxdxvvvdvvvxvdxxdv dvvvxdxvvv 22 1 vvv dvvv x dx 2 21 vvv dvvv x dx 2 21 ******************************** 22. 23 132 yx yx dx dy dxyxdyyx 13223 023132 dyyxdxyx 023132 dyyxdxyx hipótesebaba ª20111221 '''' dydydxdxkyyhxx 0'2''3'1'3'2 dykyhxdxkyhx 0'2'3'3'13'32'2 dykyhxdxkyhx 023 0132 kh kh 11 1 11 7 kh 0'''3''3'2 dyyxdxyx 0''''3''3'2 vdxdvxvxxdxvxx 0''3'32' vdxdvxvdxvx 0''3'32 vdxdvxvdxv 0'''3'3'32 2 dxvvdvxvdxdvxdxv dvvxdxvv 3''62 2 262 3 ' ' vv dvv x dx 262 3 ' ' vv dvv x dx Cvvx 262ln 2 1'ln ' ' x yv C x y x yx 2 ' ' ' '62ln 2 1'ln kyyhxx '' C x y x y x 2 11 7 11 1 11 7 11 16 2ln 2 1 11 7ln 23. 01332 dyyxdxyx 01332 dyyxdxyx hipótesebaba ª2071221 '''' dydydxdxkyyhxx 0'1''3''3'2 dykyhxdxkyhx 0'1'3'3'3'32'2 dykyhxdxkyhx 013 032 kh kh 21 6 7 3 kh 0'''3''3'2 dyyxdxyx 0''''3''3'2 vdxdvxvxxdxvxx 0''3'32' vdxdvxvdxvx 0''3'32 vdxdvxvdxv 0'3'3'''32 2 vdxdvxdxvvdvxdxv dvvxdxvv 3''62 2 262 3 ' ' vv dvv x dx 262 3 ' ' vv dvv x dx Cvvx 262ln 2 1'ln ' ' x yv C x y x yx 2 ' ' ' '62ln 2 1'ln kyyhxx '' C x y x y x 2 7 3 21 6 7 3 21 66 2ln 2 1 7 3ln 24. 05242 dyyxdxyx 05242 dyyxdxyx hipótesebaba ª2031221 '''' dydydxdxkyyhxx 0'5''2'4'2' dykyhxdxkyhx 0'5'2'2'42'2' dykyhxdxkyhx 052 042 kh kh 12 kh 0'''2''2' dyyxdxyx 0''''2''2' vdxdvxvxxdxvxx 0''2'21' vdxdvxvdxvx 0''2'21 vdxdvxvdxv 0'''2'2'21 2 dxvvdvxvdxdvxdxv dvvxdxv 2''1 2 21 2 ' ' v dvv x dx 1 2 ' ' 2v dvv x dx C v vvx 1 1ln1ln 2 1'ln 2 ' ' x yv C x y x y x yx 1 ' ' 1 ' ' ln1 ' 'ln 2 1'ln 2 kyyhxx '' C x y x y x yx 1 2 1 1 2 1 ln1 2 1ln 2 12ln 2 25. 136 12 yx yx dx dy dxyxdyyx 12136 013612 dyyxdxyx 013612 dyyxdxyx hipótesebaba ª101221 dydxbdyadxdt yxtbyaxt 2 2 dxdtdy b adxdtdy xty b axty 2 2 021236122 dxdtxtxdxxtx 02131 dxdttdxt 02631 dxdttdxtdtdxt dttdxt 3135 dt t tdx 35 31 dt t tdx 35 31 Cttx 35ln 25 4 5 3 Cyxyxx 325ln 25 42 5 3 Cyxyxx 325ln 25 42 5 3 Cyxyxx 25325ln421525 Cxyyxx 3105ln4153025 Cxyyx 3105ln4155 26. 0232132 dyyxdxyx hipótesebaba ª101221 dydxbdyadxdt yxtbyaxt 32 32 3 2 3 2 dxdtdy b adxdtdy xty b axty 0 3 22 3 2321 3 232 dxdtxtxdxxtx 0 3 221 dxdttdxt 0 3 4 3 2 3 2 3 1 dxdtdxtdttdxt 042233 dxdttdxtdtdxt dttdxt 27 dt t tdx 7 2 dt t tdx 7 2 Cttx 7ln9 Cttx 7ln9 Cyxyxx 732ln932 Cyxyx 732ln933 27. yx yx dx dy 1 331 dxyxdyyx 3311 01331 dyyxdxyx 01331 dyyxdxyx 01133 dyyxdxyx hipótesebaba ª101221 dydxbdyadxdt yxtbyaxt 33 33 3 3 3 3 dxdtdy b adxdtdy xty b axty 0 3 31 3 31 3 333 dxdtxtxdxxtx 0 3 31 3 1 dxdttdxt 0 339 1 dxdtdxtdttdxt 093399 dxdttdxtdtdxt dttdxt 3186 dtt tdx 186 3 dt t tdx 3 3 6 1 Cttx 3ln 6 1 Cyxyxx 333ln33 6 1 Cyxyxx 333ln33 6 1 Cyxyxx 2333ln233 3 12 Cyxyxx 333ln22 Cyxyx 333ln23 28. 05242 dxyxdyyx 04252 dyyxdxyx hipótesebaba ª2031221 '''' dydydxdxkyyhxx 0'4''2'5'2' dykyhxdxkyhx 0'4'2'2'52'2' dykyhxdxkyhx 042 052 kh kh 21 kh 0'''2''2' dyyxdxyx 0''''2''2' vdxdvxvxxdxvxx 0''2'21' vdxdvxvdxvx 0''2'21 vdxdvxvdxv 0'''2'2'21 2 dxvvdvxvdxdvxdxv dvvxdxv 2''1 2 21 2 ' ' v dvv x dx 1 2 ' ' 2v dvv x dx C v vvx 1 1ln1ln 2 1'ln 2 ' ' x yv C x y x y x yx 1 ' ' 1 ' ' ln1 ' 'ln 2 1'ln 2 kyyhxx '' C x y x y x yx 1 1 2 1 1 2 ln1 1 2ln 2 11ln 2 29. 342 12 yx yx dx dy dxyxdyyx 12342 034212 dyyxdxyx 034212 dyyxdxyx hipótesebaba ª101221 dydxbdyadxdt yxtbyaxt 2 2 2 2 dxdtdy b adxdtdy xty b axty 0 2 3 2 421 2 2 dxdtxtxdxxtx 0 2 321 dxdttdxt 0 2 3 2 31 dxdttdxtdtdxt 0332222 dxdttdxtdtdxt dttdxt 3254 dtt tdx 54 32 dt t tdx 54 32 Cttx 54ln 8 1 2 1 Cyxyxx 524ln 8 12 2 1 Cyxyxx 524ln 8 12 2 1 Cyxyxx 8524ln248 Cyxyxx 584ln848 Cyxyx 584ln84 30. 05634 dyyxdxyx 05634 dyyxdxyx hipótesebaba ª2021221 '''' dydydxdxkyyhxx 0'5'6''3'4' dykyhxdxkyhx 0'56'6''34'4' dykyhxdxkyhx 056 034 kh kh 11 kh 0''6'''4' dyyxdxyx 0'''6'''4' vdxdvxvxxdxvxx 0''61'41' vdxdvxvdxvx 0''61'41 vdxdvxvdxv 0'6'6'''41 2 dxvvdvxvdxdvxdxv dvvxdxvv 61''651 2 2651 61 ' ' vv dvv x dx 156 16 ' ' 2 vv dvv x dx C v v vvx 3 1 2 1 ln9156ln 2 1'ln 2 ' ' x yv C x y x y x y x yx 3 1 ' ' 2 1 ' ' ln91 ' '5 ' '6ln 2 1'ln 2 kyyhxx '' C x y x y x y x yx 3 1 1 1 2 1 1 1 ln91 1 15 1 16ln 2 11ln 2 31. 013923 dyyxdxyx hipótesebaba ª101221 dydxbdyadxdt yxtbyaxt 3 3 dtdxdy b adxdtdy txy b axty 3 3 031339233 dtdxtxxdxtxx 03132 dtdxtdxt 03932 dxdttdxtdtdxt dttdxt 13510 dt t tdx 510 13 dt t tdx 510 13 Cttx 510ln 20 1 10 3 Cyxyxx 5310ln 20 13 10 3 Cyxyxx 5310ln 20 13 10 3 Cyxyxx 205310ln3620 Cyxyxx 51030ln61820 Cyxyx 51030ln62 Equações Diferenciais Exatas – Lista de Exercícios Resolvida. 1. 0222 dyxydxyx x n y m yy 22 ydxyxyxf 22, yxyxyxf 2 3 3 , yyx y yxf '2, yyxyx '22 0' y dyy 0' Cy Cxyxyxf 2 3 3 , Cxyx 2 3 3 0 xyxC 2 3 3 xyxC 2 3 3 2. 02312 dyyxdxyx x n y m 11 ydxyxyxf 12, yxyxxyxf 2, yx y yxf ', yxyx '23 23' yy dyyy 23' Cyyy 2 2 3 2 Cyyxyxxyxf 2 2 3, 2 2 Cyyxyxx 2432220 22 yyxyxxC 432222 22 yyxyxxC 43222 22 3. 02 dyyxedxe yy x n y m yy ee ydxeyxf y , yxeyxf y , yxe y yxf y ', yxeyxe yy '2 yy 2' dyyy 2' Cyy 2 Cyxeyxf y 2, Cyxe y 20 2yxeC y 2yxeC y 4. 0cos223 dyyxydxyx x n y m yy 22 ydxyxyxf 23, yxyxyxf 2 4 4 , yyx y yxf '2, yyxyxy '2cos2 yy cos' dyyy cos' Cyy sin Cyxyxyxf sin 4 , 2 4 Cyxyx sin 4 0 2 4 yxyxC sin 4 2 4 yxyxC sin 4 2 4 5. 012coscos dy y xxyxdx x yxyy x n y m x xyxyxy x xyxyxy 1sincos1sincos ydx x yxyyyxf cos, yxyxyyxf 2sin, yxxyx y yxf '2cos, yxxyx y xxyx '2cos12cos y y 1' y dyy' Cyy ln Cyxyxyyxf ln2sin, Cyxyxy ln2sin0 yxyxyC ln2sin yxyxyC ln2sin 6. 07312 dyydxx x n y m 00 ydxxyxf 12, yxxyxf 2, y y yxf '0, yy '73 73' yy dyyy 73' Cyyy 7 2 3 2 Cyyxxyxf 7 2 3, 2 2 Cyyxx 7 2 30 2 2 yyxxC 7 2 3 22 yyxxC 7 2 3 22 7. 08445 3 dyyxdxyx x n y m 44 ydxyxyxf 45, yyxxyxf 4 2 5, 2 yx y yxf '4, yxyx '484 3 38' yy dyyy 38' Cyy 42 Cyyxxyxf 4 2 24 2 5, Cyyxx 4 2 24 2 50 4 2 24 2 5 yyxxC 4 2 24 2 5 yyxxC 8. 04232 22 dyyxdxxy x n y m yxyx 44 ydxxyyxf 32, 2 yxxyyxf 3, 22 yyx y yxf '2, 2 yyxyx '242 22 4' y dyy 4' Cyy 4 Cyxxyyxf 43, 22 Cyxxy 430 22 yxxyC 4322 yxxyC 4322 9. 0lnln43 32 dyyxdxxyx x n y m 22 33 xx ydxxyxyxf ln43, 2 yxxxyxyxf 4ln4, 3 yx y yxf ', 3 yxyx 'ln 33 yy ln' dyyy ln' Cyyyy ln Cyyyxxxyxyxf ln4ln4, 3 Cyyyxxxyx ln4ln40 3 yyyxxxyxC ln4ln43 yyyxxxyxC ln4ln43 10. 0cos23sin 223 dyxyxydxxxyy x n y m xyyxyy sin23sin23 22 ydxxxyyyxf sin, 23 yxxyxyyxf 2 cos, 2 23 yxyxy y yxf 'cos23, 2 yxyxyxyxy 'cos23cos23 22 0' y dyy 0' Cy Cxxyxyyxf 2 cos, 2 23 Cxxyxy 2 cos0 2 23 2 cos 2 23 xxyxyC 2 cos 2 23 xxyxyC 11. xy xy xey ye dx dy 2 2 dxyedyxey xyxy 22 022 dyxeydxye xyxy x n y m ydxyeyxf xy 2, yexyxf xy 2, yxe y yxf xy ', yxexey xyxy '2 yy 2' dyyy 2' Cyy 2 Cyexyxf xy 22, Cyex xy 220 22 yexC xy 22 yexC xy 12. 03154 2423 dyxyxdxyxyx x n y m 1414 33 xx ydxyxyxyxf 23 154, yyxxyxyxf 34 5, yxx y yxf ', 4 yxxxyx '3 424 23' yy dyyy 23' Cyy 3 Cyyxxyxyxf 334 5, Cyyxxyx 334 50 334 5 yyxxyxC 334 5 yyxxyxC 13. 11,012 22 ydyxxydxyx 0122 222 dyxxydxyxyx x n y m yxyx 2222 ydxyxyxyxf 22 2, yxyyxxyxf 22 3 3 , yyxx y yxf '2, 2 yyxxxxy '212 22 1' y dyy' Cyy Cyxyyxxyxf 22 3 3 , Cyxyyxx 22 3 3 0 yxyyxxC 22 3 3 yxyyxxC 22 3 3 11111 3 1 223 C 3 4 C yxyyxx 22 3 33 4 14. 21,0146524 ydyxydxxy x n y m 44 ydxxyyxf 524, yxxxyyxf 54, 2 yx y yxf '4, yxxy '4146 16' yy dyyy 16' Cyyy 23 Cyyxxxyyxf 22 354, Cyyxxxy 22 3540 yyxxxyC 22 354 yyxxxyC 22 354 yyxxxyC 22 354 223151214 22 C 8C yyxxxy 22 3548 15. 03131 dyx y dxy x x n y m 11 ydxy x yxf 31, yyxxxyxf ln3, yx y yxf ', yxx y '31 y y 31' dy y y 31' Cyyy ln3 Cyyyxxxyxf ln3ln3, Cyyyxxx ln3ln30 yxyyxxC lnln3 xyyyxxC ln3 16. 0 91 1 23 2 32 yx dy dx x yx dyyxdx x yx 232 32 91 1 0 91 1 23 2 32 dyyxdx x yx x n y m 2222 33 yxyx ydx x yxyxf 2 32 91 1, yxyxyxf 3arctan 3 1 3 , 33 yyx y yxf ', 23 yyxyx '2323 0' y dyy 0' Cy Cxyxyxf 3arctan 3 1 3 , 33 Cxyx 3arctan 3 1 3 0 33 xyxC 3arctan3 33 xyxC 3arctan33 17. 0coscossinsintan dyyxdxyxx x n y m yxyx cossincossin ydxyxxyxf sinsintan, yyxxyxf sincossecln, yyx y yxf 'coscos, yyxyx 'coscoscoscos 0' y dyy 0' Cy Cyxxyxf sincossecln, Cyxx sincossecln0 yxxC sincossecln yxxC sincossecln 18. xyx dx dyyx 44221 32 dxxyxdyyx 44221 32 022144 23 dyyxdxxyx x n y m xx 44 ydxxyxyxf 44, 3 yyxxyxf 24 2, yx y yxf '2, 2 yxyx '2221 22 12' yy dyyy 12' Cyyy 2 Cyyyxxyxf 224 2, Cyyyxx 224 20 yyyxxC 224 2 19. eydyyxxydxxyxxy 0,0lnsin223cos 322 x n y m 22 3cos23cos2 xxyxxy ydxxyxxyyxf 23cos, 22 yxyxxyyxf 232 sin, yxxy y yxf 'sin2, 3 yxxyyxxy 'sin2lnsin2 33 yy ln' dyyy ln' Cyyyy ln Cyyyxyxxyyxf lnsin, 232 Cyyyxyxxy lnsin0 232 yyyxyxxyC lnsin 232 yyyxyxxyC lnsin 232 eeeeeC ln000sin 232 0C yyyxyxxy lnsin0 232 20. 262 xyxe dx dyx x dxxyxedyx x 262 062 2 dyxdxxyxex x n y m 11 ydxxyxeyxf x 262, yxyxexeyxf xx 3222, yx y yxf ', yxx ' 0' y dyy 0' Cy Cxyxexeyxf xx 3222, Cxyxexe xx 32220 3222 xyxexeC xx 21. 012 dyxydxy x n y m yy 2 :hipótesesegundadaAtravés yy yy m y m x n 12 2 y eeyxI y dy y 1, ln 1 011 2 dyxydxy y 01 dyyxdxy ** x n y m 11 ydxyyxf , yyxyxf , yx y yxf ', yxyx '1 1' yy dyyy 1' Cyy ln Cyyxyxf ln, Cyyx ln0 yyxC ln yyxC ln 22. 0222 dyxydxyx x n y m yy 22 :hipóteseprimeiradaAtravés xxy yy n x n y m 2 2 22 2 ln2 2 1, x eeyxI x dx x 021 222 dyxydxyxx 021 2 2 dy x ydx x y ** x n y m 22 22 x y x y ydx x yyxf 2 21, y x yxyxf 2 , y x y y yxf '2, y x y x y '22 0' y dyy 0' Cy C x yxyxf 2 , C x yx 2 0 2 x yxC 2 x yxC 23. dxexdxydyx x2 02 dyxdxydxex x 02 dyxdxyex x x n y m 11 :hipóteseprimeiradaAtravés xxn x n y m 211 2 ln2 2 1, x eeyxI x dx x 01 22 dyxdxyexx x 012 dy x dx x yex ** x n y m 22 11 xx ydx x yeyxf x 2, y x yeyxf x , y xy yxf '1, y xx '11 0' y dyy 0' Cy C x yeyxf x , C x yex 0 Cxyxex 0 Cxxey x 24. 02 dyxdxydyy 02 dyxydxy x n y m 11 :hipótesesegundadaAtravés yym y m x n 211 2 ln2 2 1, y eeyxI y dy y 01 22 dyxydxyy 011 2 dy y xdx y ** x n y m ydx y yxf 1, y y xyxf , y y x y yxf ', 2 y y x y x '1 22 1' y dyy 1' Cyy Cy y xyxf , Cy y x 0 2yxCy 2yxCy 25. 0ln3 dyxydx x y x n y m xx 11 ydx x yyxf , yxyyxf ln, yx y yxf 'ln, yxxy 'lnln3 3' yy dyyy 3' Cyy 4 4 Cyxyyxf 4 ln, 4 Cyxy 4 ln0 4 4ln44 yxyC 4ln4 yxyC 26. 0ln dyxxdxyx x n y m 1ln1 x :hipóteseprimeiradaAtravés xxx x n x n y m 1 ln 1ln1 x eeyxI x dx x 1, ln 1 0ln1 dyxxdxyx x 0ln1 dyxdx x y ** x n y m xx 11 ydx x yyxf 1, yxyxyxf ln, yx y yxf 'ln, yxx 'lnln 0' y dyy 0' Cy Cxyxyxf ln, Cxyx ln0 27. 02 3 dyxdxxy x n y m 12 :hipóteseprimeiradaAtravés xxn x n y m 112 xeeyxI x dx x ln 1 , 02 3 dyxdxxyx 02 24 dyxdxxxy ** x n y m ydxxxyyxf 42, yxyxyxf 5 , 5 2 yx y yxf ', 2 yxx '22 0' y dyy 0' Cy Cxyxyxf 5 , 5 2 Cxyx 5 0 5 2 5 5 2 xyxC 5 5 2 xyxC 28. 0343 322 dyyxdxyx x n y m yxyx 22 126 :hipótesesegundadaAtravés yyx yxyx m y m x n 2 3 612 22 22 2ln2 2 , yeeyxI y dy y 0343 3222 dyyxdxyxy 0343 23342 dyyyxdxyx ** x n y m 3232 1212 yxyx ydxyxyxf 423, yyxyxf 43, yyx y yxf '4, 33 yyxyyx '4124 33233 212' yy dyyy 212' Cyy 34 Cyyxyxf 343 4, Cyyx 343 40 343 4yyxC 343 4yyxC 29. 022 dyxydxxyx x n y m yy 2 :hipóteseprimeiradaAtravés xxy yy n x n y m 12 xeeyxI x dx x ln 1 , 022 dyxydxxyxx 02223 dyyxdxxxyx ** x n y m xyxy 22 ydxxxyxyxf 223, yxyxxyxf 324 , 3224 yyx y yxf ', 2 yyxyx '22 0' y dyy 0' Cy Cxyxxyxf 324 , 3224 Cxyxx 324 0 3224 Cxyxx 124630 3224 3224 46312 xyxxC 3224 463 xyxxC 30. 0344 dyxydxyx x n y m 334 yy :hipóteseprimeiradaAtravés xxy yy n x n y m 54 3 33 5 ln5 5 1, x eeyxI x dx x 01 3445 dyxydxyxx 01 4 3 5 4 dy x ydx x y x ** x n y m 5 3 5 3 44 x y x y ydx x y x yxf 5 41, y x yxyxf 4 4 4 ln, y x y y yxf ', 4 3 y x y x y '4 3 4 3 0' y dyy 0' Cy C x yxyxf 4 4 4 ln, C x yx 4 4 4 ln0 444 4ln40 Cxyxx 444 ln4 Cxxxy Equações diferenciais lineares. Lista de exercícios resolvida. 1. 2 x x y dx dy 21 xxQ x xP dxxQeeCey dxxPdxxPdxxP dxxeeCey dx x dx x dx x 2 111 dxxeeCey xxx 2lnlnln dxxxxCxy 2 1 xxxCxy ln2 Cxxxy ln2 2. xxy dx dy sintan xxQxxPsintan dxxQeeCey dxxPdxxPdxxP dxxeeCey dxxdxxdxx sin tantantan dxxeeCey xxx sincoslnseclnsecln dxxxxxCy sincossecsec 2 sinsecsec 2 xxxCy Cxxy 2 sinsec 2 3. 0cot x x x y dx dy x x x y dx dy cot x xxQ x xP cot1 dxxQeeCey dxxPdxxPdxxP dx x xeeCey dx x dx x dx x cot 111 dx x xeeCey xxx cotlnlnln dxx xx xx Cy cot1 x xx Cy sinln1 x Cx y sinln 4. 0cos1sin dxydyyx dxydyyx cos1sin dy dx y yx cos 1sin yy y y x dy dx cos 1 cos sin cos yyyx dy dx sectansec yyyQyyP sectansec dyyQeeCex dyyPdyyPdyyP dyyyeeCex dyydyydyy sectan secsecsec dyyyeeCex yyyyyy sectantanseclntanseclntansecln dyyy yy yyyyCx sectantansec 1tansectansec dy yy y y y y yyyyCx cos 1 cos sin cos sin cos 1 1tansectansec dy y y y yyyyyCx cos 1sin 1sin costansectansec dy y yyyyyCx 1sin 1sintansectansec dy yy yyyyyCx 1sin 1 1sin sintansectansec dyy y y dy y y y yyyyyCx sin1 sin1 sin1 1 sin1 sin1 sin1 sintansectansec yyy y yyyyCx tantan cos 2tansectansec yyyyyyyCx tan2sec2tansectansec Cyyyyyx tan2sec2tansec 5. xy dx dyx arctan1 2 22 1 arctan 1 x x x y dx dy 22 1 arctan 1 1 x xxQ x xP dxxQeeCey dxxPdxxPdxxP dx x xeeCey x dx x dx x dx 2 111 1 arctan222 dx x xeeCey xxx 2 arctanarctanarctan 1 arctan xxxx exeeCey arctanarctanarctanarctan arctan 1arctanarctan xCey x 6. y dx dy 5 05 y dx dy 05 xQxP dxxQeeCey dxxPdxxPdxxP dxeeCey dxdxdx 0 555 05 xCey xCey 5 7. 4123 y dx dy 3 44 y dx dy 3 44 xQxP dxxQeeCey dxxPdxxPdxxP dxeeCey dxdxdx 3 4444 dxeeCey xxx 3 4444 xxx eeCey 444 3 1 3 14 xCey 8. xey dx dy 3 xexQxP 31 dxxQeeCey dxxPdxxPdxxP dxeeeCey x dxdxdx 3111 dxeeeCey xxxx 3 dxeeCey xxx 4 xxx eeCey 4 4 1 xx eCey 3 4 1 9. 223' xyxy 223 xyx dx dy 223 xxQxxP dxxQeeCey dxxPdxxPdxxP dxxeeCey dxxdxxdxx 2333 222 dxxeeCey xxx 2 333 333 3 1 xxx eeCey 3 13 xCey 10. 1'2 xyyx 12 xy dx dyx 22 1 xx xy dx dy 2 1 xx y dx dy 2 11 x xQ x xP dxxQeeCey dxxPdxxPdxxP dx x eeCey dx x dx x dx x 2 111 1 dx x eeCey xxx 2 lnlnln 1 dxxxxxCy 2 111 x xx Cy ln1 xC x y ln1 11. 024 2 dxydyyx dxydyyx 24 2 dy dx y yx 2 4 2 y yx dy dx 2 4 2 y y x dy dx 2 2 y y x dy dx 2 2 yyQ y yP 2 2 1 dyyQeeCex dyyPdyyPdyyP dyyeeCex dy y dy y dy y 22 1 2 1 2 1 dyyeeCex yyy 2 ln 2 1 ln 2 1 ln 2 1 dyyeeCex yyy 2ln 1ln1ln dyyyyyCx 2 11 5 411 5 y yy Cx 5 4 4 y y Cx 5 4 2y y Cx 12. dxyxxdyx sin x yxx dx dy sin x yx dx dy sin x x y dx dy sin xxQ x xP sin1 dxxQeeCey dxxPdxxPdxxP dxxeeCey dx x dx x dx x sin 111 dxxeeCey xxx sinlnlnln dxxxxxCy sin 11 xxx xx Cy sincos1 x xx x Cy sincos x x xCy cossin 13. 01 ye dx dye xx xx x ee ye dx dy 1 0 1 01 x x e ye dx dy 01 xQe exP x x dxxQeeCey dxxPdxxPdxxP dxeeCey dx e edx e edx e e x x x x x x 0111 01ln xeCey xe Cy 1 1 xe Cy 1 14. 1sincos xy dx dyx xx xy dx dy cos 1 cos sin xxy dx dy sectan xxQxxP sectan dxxQeeCey dxxPdxxPdxxP dxxeeCey dxxdxxdxx sec tantantan dxxeeCey xxx secseclncoslncosln dxxxxxCy secseccoscos dxxxxCy 2seccoscos xxxCy tancoscos x xxxCy cos sincoscos xxCy sincos 15. xxy dx dyx 34 x xx x y dx dy 34 14 2 x x y dx dy 14 2 xxQ x xP dxxQeeCey dxxPdxxPdxxP dxxeeCey dx x dx x dx x 12 444 dxxeeCey xxx 12ln4ln4ln4 dxxxxxCy 1 11 24 44 57 1 57 44 xx xx Cy 57 3 4 xx x Cy 16. xeyxx dx dyx 22 22 2 x e x yxx dx dy x 2 21 x ey xdx dy x 2 21 x exQ x xP x dxxQeeCey dxxPdxxPdxxP dx x eeeCey xdx x dx x dx x 2 212121 dx x eeeCey x xxxxxx 2 ln2ln2ln2 dx x eeeeeeCey x xxxxxx 2 ln2ln2ln2 dx x exe xexe Cy x x xx 2 2 22 11 dxexexe Cy xxx 2 22 1 x xx exexe Cy 222 2 11 x x x ex e xe Cy 222 2 22 2x e xe Cy x x 17. 01cossincos 32 dxxydyxx dxxydyxx 1cossincos 32 xx xy dx dy sincos 1cos 2 3 xx y x x dx dy sincos 1 sin cos 2 xxyx dx dy seccosseccot 2 xxxQxxP seccosseccot 2 dxxQeeCey dxxPdxxPdxxP dxxxeeCey dxxdxxdxx seccossec2 cotcotcot dxxxeeCey xxx seccossec2sinlnsinlnsinln dxxxxxxCy seccossecsinsin 1 sin 1 2 dxxxx Cy 2sec sin 1 sin x xx Cy tan sin 1 sin x x xx Cy cos sin sin 1 sin xx Cy cos 1 sin xxCy secseccos 18. 02 dyyexxydxy y dyyexxydxy y 2 y yexxy dy dx y 2 ye y xx dy dx 2 ye y x dy dx 21 yeyQ y yP 21 dyyQeeCex dyyPdyyPdyyP dyeeeCex y dy y dy y dy y 212121 dyeeeCex yyyyyyy ln2ln2ln2 dyeeeeeeCex yyyyyyy ln2ln2ln2 dyeyeyeyeCx yy yy 2 22 11 dyyeyeye Cx yyy 22 22 1 y yy yy e yeye yeye Cx 2 222 22 4 1 22 1 22 422 y e y ee ye Cx yyy y 19. xeyx dx dyx 313 x ey x x dx dy x313 x ey xdx dy x313 x exQ x xP x313 dxxQeeCey dxxPdxxPdxxP dx x eeeCey xdx x dx x dx x 3131313 dx x eeeCey x xxxxxx 3 ln3ln3ln3 dx x eeeeeeCey x xxxxxx 3 ln3ln3ln3 dx x exe xexe Cy x x xx 3 3 33 11 dxxexe Cy xx 33 1 x xexe Cy xx 33 1 xx exe Cy 33 1 113 x C e y x 20. 04 6 dyyxdxy dyyxdxy 64 y yx dy dx 64 544 y y x dy dx 544 y y x dy dx 544 yyQ y yP dyyQeeCex dyyPdyyPdyyP dyyeeCex dy y dy y dy y 5 444 4 dyyeeCex yyy 5ln4ln4ln4 4 dyyyyCyx 5 4 44 41 dyyyCyx 444 2 4 244 yyCyx 64 2yCyx 21. xx x ee ey dx dy 21 xx x ee exQxP 211 dxxQeeCey dxxPdxxPdxxP dx ee eeeCey xx x dxdxdx 2 111 1 dx ee eeeCey xx x xxx 21 dx e e eeeCey x x x xxx 1 1 2 dx e e eeeCey x x x xxx 1 1 2 2 dx e eeeeCey x xx xxx 1 1 2 2 dx e eeCey x x xx 1 1 2 2 Professor, não conseguimos resolver esta integral! 22. 022 2 dyyxyxdxy dyyxyxdxy 22 2 y yxyx dy dx 22 2 22 xy y x dy dx 221 y y x dy dx 221 yQy y yP dyyQeeCex dyyPdyyPdyyP dyeeCex dyy y dyy y dyy y 2 212121 dyeeCex yyyyyy 2 222 lnlnln dyeeeeeCex yyyyyy 2 222 lnlnln dyyeyeye Cx y yy 211 2 22 2 22 1 y yy e yeye Cx yye Cx y 1 2 11 2ye C y x 23. cossec r d dr cossec r d dr cossec QP dQeeCer dPdPdP deeCer ddd cos secsecsec deeCer costanseclntanseclntansecln dCr costansectansec 1 tansec 1 d Cr sin1 tansec 1 tansec cos tansec 1 tansec Cr cos tansec 1 Cr costansec Cr 24. xyy dx dyx 4852 2 22 485 x xyy dx dy 22 2 5 2 48 x y x x dx dy 22 2 5 2 48 x xQ x xxP dxxQeeCey dxxPdxxPdxxP dx x eeCey dx x xdx x xdx x x 2 2 48 2 48 2 48 2 5222 dx x eeCey xxx 2 2ln22ln22ln2 2 5222 dx x x xx Cy 2 22 2222 2 52 2 1 2 1 dxxxx Cy 244 252 1 2 3 25 2 1 2 3 44 x xx Cy 23 5 2 4 xx Cy 63 5 2 4 xx Cy 25. 20205 ycomy dx dy 205 xQxP dxxQeeCey dxxPdxxPdxxP dxeeCey dxdxdx 20 555 dxeeCey xxx 20555 xxx eeCey 555 4 45 xCey 42 05 Ce 42 C 2C 42 5 xey 26. 00,,, iicomcEeRLsendoERidt diL tes L Ei L R dt di L EtQ L RtP dttQeeCei dttPdttPdttP dt L EeeCei dt L Rdt L Rdt L R dt L EeeCei L Rt L Rt L Rt L Rt L Rt L Rt e R EeCei R ECei L Rt R ECei L R 0 0 R EiC 0 R Ee R Eii L Rt 0 27. 10,costan' 2 ycomxyxy xyx dx dy 2costan xxQxxP 2costan dxxQeeCey dxxPdxxPdxxP dxxeeCey dxxdxxdxx 2tantantan cos dxxeeCey xxx 2seclncoslncosln cos dxxxxxCy 2cosseccoscos dxxxxCy coscoscos xxxCy sincoscos 0sin0cos0cos1 C 1C 1sincos xxy 28. 2000,50 TcomTk dt dT kkT dt dT 50 kkT dt dT 50 ktQktP 50 dttQeeCeT dttPdttPdttP dtkeeCeT dtkdtkdtk 50 dtkeeCeT ktktkt 50 ktktkt eeCeT 50 50 ktCeT 50200 0 kCe 50200 C 150C 50150 kteT 29. 101,ln1 ysendoxy dx dyx 1 ln 1 x x x y dx dy 1 ln 1 1 x x xQ x xP dxxQeeCey dxxPdxxPdxxP dx xx eeCey dx x dx x dx x 1 ln1 1 1 1 1 1 dx x x eeCey xxx 1 ln1ln1ln1ln dx x x x xx Cy 1 ln 1 1 1 1 1 dxxxx Cy ln 1 1 1 xxx xx Cy ln 1 1 1 xxxC x y ln 1 1 11ln1 11 110 C 21C xxx x y ln21 1 1 xxxyx ln211 30. 63,022 ycomy dx dyxx 2 0 2 2 xxxx y dx dy 0 2 2 xx y dx dy 0 2 2 xQ xx xP dxxQeeCey dxxPdxxPdxxP dxeeCey dx xx dx xx dx xx 02 2 2 2 2 2 2 ln x x Cey 2x xCy 23 36 C 2C 2 2 x xy 2 2 x xy 31. 25, ysendo xy y dx dy dxydyxy dy dx y xy y xy dy dx y x dy dx 1 1 y x dy dx 11 yQ y yP dyyQeeCex dyyPdyyPdyyP dyeeCex dy y dy y dy y 1 111 dyeeCex yyy lnlnln dyyyyCx 11 2 1 2y yy Cx 2 y y Cx 2 2 2 5 C 8C 2 8 y y x Equações de Bernoulli. Lista de exercícios resolvida. 1. 232 xy x y dx dy xx xPnxP 22211* xxxQnxQ 33211* dxxQeeCev dxxPdxxPdxxP **** 1211 yyyv n dxxeeCey dx x dx x dx x 3 222 1 dxxeeCey xxx 3ln2ln2ln21 dxxxxxCy 3 11 2 22 1 4 31 4 22 1 x xx Cy 2 4 2 1 4 3 x x x Cy 2 4 4 341 x xC y 42 344 xCyx 4 2 3 4 xC xy 2. 32 xyxy dx dy xxxPnxP 42311* xxxQnxQ 2311* dxxQeeCev dxxPdxxPdxxP **** 2311 yyyv n dxxeeCey dxxdxxdxx 2 4442 dxxeeCey xxx 2 222 2222 222 2222 2 1 xxx eeCey 2 1 22 2 xe Cy 2 2 2 2 2 2 21 x x e eC y 22 222 22 xx eCye 2 2 2 2 2 2 x x eC ey 3. 33 yxxy dx dy xxxPnxP 2311* 33 2311* xxxQnxQ dxxQeeCev dxxPdxxPdxxP **** 2311 yyyv n dxxeeCey dxxdxxdxx 32222 2 dxxeeCey xxx 32 2 222 2222 22 xxxx eexeCey 122 2 xCey x 11 22 2 xCe y x 11 22 2 xCey x 1 1 2 2 2 xCe y x 1 1 22 xCe y x 212 1 1 2 xCe y x 212 12 xCey x 4. 2 1 y y dx dyx 2 1 xyx y dx dy xx xPnxP 31211* xx xQnxQ 31211* dx x eeCey dx x dx x dx x 3 333 3 dx x eeCey xxx 3ln3ln3ln33 dx x x xx Cy 311 333 3 3 33 3 11 x xx Cy 13 3 x Cy 3 3 3 x xCy Cxyx 333 5. 13 xyy dx dy yxy dx dy 4 4xyy dx dy 31411* xPnxP xxxQnxQ 3411* 3411 yyyv n dxxeeCey dxdxdx 3 3333 dxxeeCey xxx 33333 xxxx exeeCey 33333 3 1 3 133 xCey x 3 11 3 3 xCey x dxxQeeCev dxxPdxxPdxxP **** 3211 yyyv n dxxQeeCev dxxPdxxPdxxP **** xCex y 3 3 3 11 6. xyy dx dyx 22 x y x y dx dy 2 2 2 2 x y x y dx dy xx xPnxP 11211* 22 11211* xx xQnxQ 1211 yyyv n dx x eeCey dx x dx x dx x 2 111 1 1 dx x eeCey xxx 2 lnlnln1 1 x xx Cy ln111 x xC y ln1 xC y x ln Cx y x ln 7. 2 11,32 42 ycomyxy dx dyx 2 432 x y x y dx dy xx xPnxP 62411* 22 93411* xx xQnxQ dxxQeeCev dxxPdxxPdxxP **** dx x eeCey dx x dx x dx x 2 666 3 9 dx x eeCey xxx 2 ln6ln6ln63 9 dx x x xx Cy 2 6 66 3 911 5 911 5 66 3 x xx Cy xx Cy 5 91 6 3 x yxC 5 936 15 9 2 11 3 6C 5 49 C xx y 5 91 5 49 6 3 xx y 5 9 5 49 6 3 8. 33 yxy dx dyx 32 yx x y dx dy xx xPnxP 21311* 22 2311* xxxQnxQ 2311 yyyv n dxxeeCey dx x dx x dx x 2 222 2 2 dxxeeCey xxx 2ln2ln2ln22 2 dxxxxCxy 2 2 222 21 dxxQeeCev dxxPdxxPdxxP **** 3411 yyyv n dxxQeeCev dxxPdxxPdxxP * *** xxCxy 2222 322 2xCxy 32 2 2 1 xCx y 322 21 xCxy 2322 21 yxyCx 9. yxy xdx dy 4 yxy xdx dy 4 xx xPnxP 24 2 111* xxxQnxQ 2 1 2 111* 2 1 2 111 yyyv n dxxeeCey dx x dx x dx x 2 222 2 1 dxxeeCey xxx 2 ln2ln2ln22 1 dxx x xCxy 2 1 2 222 1 xxCxy ln 2 12221 Cxxy ln 2 1221 2 2 ln 2 1 Cxxy 2 4 ln 2 1 Cxxy 10. 02 2 xy dx dyxy 0 2 1 2 yx y dx dy dxxQeeCev dxxPdxxPdxxP **** yx y dx dy 2 1 2 xx xPnxP 1 2 1111* 1 2 1111* xQnxQ 2111 yyyv n dxeeCey dx x dx x dx x 1 111 2 dxeeCey xxx 1lnlnln2 dxxxCxy 1 12 xxCxy ln2 xCxy ln2 xC x y ln 2 Cx x y ln 2 11. 222 y x y dx dy xx xPnxP 22211* 22211* xQnxQ 1211 yyyv n dxeeCey dx x dx x dx x 2 222 1 dxeeCey xxx 2ln2ln2ln21 dxxxxCy 2 1 2 221 x xxCy 2221 xCx y 21 2 dxxQeeCev dxxPdxxPdxxP **** dxxQeeCev dxxPdxxPdxxP **** xCxy 21 2 122 xyyCx 12. dxyydyx 12 x yy dx dy 12 x y x y dx dy 3 xx xPnxP 21311* xx xQnxQ 21311* 2311 yyyv n dx x eeCey dx x dx x dx x 2 222 2 dx x eeCey xxx 2ln2ln2ln22 dx x x xx Cy 211 222 2 2222 1 x xx Cy 11 22 x C y 2 2 2 1 x xC y 2 2 2 xC y x Cx y x 22 2 13. 221 xyxy dx dyx 2 2 2 11 x xy x xy dx dy dxxQeeCev dxxPdxxPdxxP **** 2 2 2 11 x xy x xy dx dy 22 11211* x x x xxPnxP 22 11211* x x x xxQnxQ 1211 yyyv n dx x xeeCey dx x xdx x xdx x x 2 1111 1 222 dx x xeeCey xxx 2 1ln 2 11ln 2 11ln 2 1 1 1 222 dx x xxxxCy 2 2 122 122 121 1 111 dxxxxxCy 2 322 122 121 111 2 122 122 121 111 xxxCy 111 212 xC y 111 2 12xCy 11 1 2 12 xC y dxxQeeCev dxxPdxxPdxxP ****
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