Exercícios Selecionados Livro Calc2 Pt5-2021 2

168 e CHAPTER 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS EX E RC | S E S 4 . 7 Answers to selected odd-numbered problems begin on page ANS-5. In Problems 1-18 solve the given differential equation. Gs) xy" — 3xy’ + 13y =4 +4 3x x2y" — 2y = 0 2. 4x2y" + y =0 36. x3y"” — 3x*y" + 6xy’ — 6y = 3 + Inx? xy" +y' =0 4, xy" — 3y' =0 In Problems 37 and 38 solve the given initial-value problem xy" + xy’ + 4y =0 6. xy" + 5xy’ + 3y =0 on the interval (—°°, 0). x°y" — 3xy’ — 2y = 0 8. x°y"” + 3xy’ — 4y =0 37. 4x°y"+y=0, y(-D)=2y(-) =4 25x7y" + 25xy’ +y=0 10. 4x?y" + 4xy’ -y =0 38. x?y" — 4xy’ + 6by =0, y(—2) = 8, y'(—2) =0 WIf xy” + 5xy’ + 4y =0 12. x?y" + 8xy’ + 6y = 0 +» , x» , Discussion Problems 13. 3x°y" + 6xy’ + y =0 14, xy” — Txy’ + 4ly =0 39. How would you use the method of this section to solve 15. x*y"” — 6y =0 16. x°y" + xy’ -—y=0 + 2)°y" + (x + 2)y' + y = 0? 18. xty + 6x3y" + Ox2y" + 3xy! + y = 0 Carry out your ideas. State an interval over which the solution is defined. In Problems 19—24 solve the given differential equation by 40. Can a Cauchy-Euler differential equation of lowest variation of parameters. order with real coefficients be found if it is known that \ , , 4 2 and | — i are roots of its auxiliary equation? Carry AX xy" — Ay = x out your ideas. 2." ’ — +2 _ 2x-y" + Sxy' +y =x — x 41. The initial-conditions y(O) = yo, y’(0) = y; apply to xy" — xy’ + y = 2x 22. x*y” — 2xy’ + 2y = x4e* each of the following differential equations: 1 2," — RH xy" bay’ ys tnx 24 ay" bay’ — y= wo * xy" — Ixy’ + 2y = 0, In Problems 25-30 solve the given initial-value problem. xy" — Axy! + by = 0. Use a graphing utility to graph the solution curve. For what values of yo and y; does each initial-value b xy" + 3xy'=0, yd) =0,y’d) =4 problem have a solution? xy" — 5xy’ + 8y = 0, y(2) = 32, y'(2) = 0 42. What are the x-intercepts of the solution curve shown 5 in Figure 4.7.1? How many x-intercepts are there for xy" +xy'’+y=0, yd)=1,y'd)=2 0<x<0 28) x*y" — 3xy' + 4y=0, yl) =5,y') =3 Q9.)xy" ty =x, y)=Ly()=- Computer Lab Assignments In Problems 43-46 solve the given differential equation by 2," __ ’ _ 6 1) _ (1) — 30. xy Sxy' + 8y = 8x, y (3) 0, (3) 0 using a CAS to find the (approximate) roots of the auxiliary ae 1 equation. In Problems 31-36 use the substitution x = e’ to transform the given Cauchy-Euler equation to a differential equation 43. 2x3y" — 10.98x2y" + 8.5xy’ + 1.3y =0 with constant coefficients. Solve the original equation by solving the new equation using the procedures in 44, x3y" + 4x7y" + Sxy’ — 9y = 0 Sections 4.3—4.5. 45. x4y + 6x3y" + 3x2y" — 3xy’ + 4y = 0 2," ri _— ( xy" + Oxy! — 20y =0 46. x4y — 6x3y" + 33x2y" — 105xy’ + 169y = 0 2. , _ x’y" — 9xy! + 25y = 0 47. Solve x*y" — xy" — 2xy' + 6y = x? by variation of (33) xy" + 1Oxy’ + 8y = x? parameters. Use a CAS as an aid in computing roots of the auxiliary equation and the determinants given in 34, x?y" — 4xy’ + 6y = Inx? (10) of Section 4.6. ANSWERS FOR SELECTED ODD-NUMBERED PROBLEMS e ANS-5 — 5 5. 6x 1 —6x 35. y= ag ~ ase * + Gxe® 35. y=cje * + c9e* — 6 — x Xx 37. y= e-*— xe 37. y=c, t+ coe * + 3x 39. y=0 39. y= ce + oxe*+3x41 41. y= (1 — =) eo V3" 4 (1 +4 =) eV3e. 4H. y=cptoxt+oe% t+ = x4 — Sx + 8x? 43. y=cje** + ce + sxet 5 _ y = cosh V3x + 7 sinh V3x 45. y=cye * + ce — e* +3 47. y =c,cos5x + cy) sin 5x + 7 sin x 49, y=cye * + Exe * — pre + sae _ —x 1 1 x Lx EXERCISES 4.4 (PAGE 148) Sl. y= ce * + ce + xe’ — gxre’ + zxet — 5 L. y=cye® + pe 2* +3 53. y = e*(c,cos 2x + cysin 2x) + fe sin x 3. y=ce* + cxxe* + Sx +3 55. y = c,cos 5x + cysin 5x — 2x cos 5x + _— —2x —2x 2 7 V3 V3 oc 5. y=cje *+ axe *+x 4x + 5 57. y= (6, cos—— x + c) sin 3.) tu 7. y = c, cos V3x + c) sin V3x + (—4x2 + 4x - 4) 038 2 2 [au + sinx + 2 cos x — x cos x < 9. y=c, + c9e* + 3x s no 7g 4 = 11. y = cye? + coxet? + 12 + Fx? er? 59. y = cy + eax + c3e + g5gX° + aX” — 76% Uv _ x 1 13. y = c, cos 2x + cy sin 2x — ux cos 2x 61. y = cyet + cyxe* + exes + Grek + x — 13 o _ X Xx 1 Xx 1 15. y = c,cosx + c)sinx — $x° cosx + $x sinx 63. y = c) + ex + ce + cyxe* + yx e+ 5° > _ 5-81 4 58% _ 1 17. y = cye*cos 2x + cye*sin 2x + } xe" sin 2x 65. y =e + ge — | = — _ 41 4 41 sy _ 1 9 19, y=cye* + Exe* — + cos x 67. y = ~y55 + jag" — GX t+ 95% O + 2 sin 2x — 2 cos 2x 69. y = —mcosx — tsinx — cos 2x + 2x cos x ie = x 3 2x oj 1 3 3 21. y =, tox + ce — 42 — Scosx + Lsinx 71. y = 2e** cos 2x — Ge* sin2x + gx + Fx? + 35K a 23. y = cye" + cyxe* + c3xe* — x — 3 — Fx et ai 25. y = cy cosx + co sin x + c3x cos x + c4x sin x EXERCISES 4.6 (PAGE 161) S Fat = 2x = 3 1. y =c, cosx + cy sinx + xsinx + cos x In| cos x| 5 27. y= V2 sin 2x ~ 3 3 = o cosx + ysinx — ! cos x QO 29, y = —200 + 200e*/5 — 3x? + 30x ea zee a 31. y = —10e72* cos x + 9e~2* sin x + Te 5. y = c, cOSx + c)sinx + 5 — ¢cos 2x O _ yg lig 33 _ _ Fy 7.y= ce + oe* + 5x sinh x a 7e Tp net 3 tcos wt At kK ° ° 1 9 y= ce + oe + i(e In|x| — e7>* =a) i 35. y = 11 — Ile + 9xe* + 2x — 12x°e* + 5e* xo t a 37. y = 6 cos x — 6(cot 1) sinx + x7 — 1 Xo > O o _ —4 sin V3x 11. y=cye* + coe * + (@* +e **) In(1 + e*) O 39. y= Sin V3 + V3c0s V3 + 2x 13. y=cje *+oe*-—e sine’ an 5 1 15. y=cje'+ ote! + iPe! Int — +Pet co cos 2x + ?sin2x + ;sinx, OSx< 7/2 . ' : — 41. y=}, 5: 17. y = cye*sinx + c,e* cos x + 4xe* sin x = 3 cos 2x + 2 sin 2x, x> 7/2 4 Los in | w 3 e*cos x In| cos x 19 y = fer? 4 Fer? + Pye? — Iyer? < . 4 4 8 4 EXERCISES 4.5 (PAGE 156) 21. y = tet + Ber — fer 4 te 1. GD — 2)3D + 2)y = sin x 23. y=cyx ! cosx + cox? sin x + x71? 3. (D— 6)(D + 2)y=x-6 _ ~yre 5. DD + 5)2 . 25. y = c, + c) cosx + c3sin. x — In|cos x| . y=e ; 7. (D —1)(D —2)(D + 5)y = xe? — sin x In|sec x + tan x| 9. D(D + 2)(D? — 2D + 4)y =4 15. D* 17. D(D — 2) 19. D244 21. D3(D? + 16) EXERCISES 4.7 (PAGE 168) 23. (D + 1)(D — 1)3 25. D(D* — 2D + 5) 1. y=cyx ! + epx? 27. 1, x, x7, x3, x4 29, 0%, e 3x2 3. y=c, +coInx 31. cos V5x, sin V5x 33. 1, e, xe* 5. y = c, cos(2 In x) + c2 sin(2 In x) ANS-6 e ANSWERS FOR SELECTED ODD-NUMBERED PROBLEMS y= C1x2-V9) 4 ¢,x2+V6) 19. x = —6cje ' — 3cxe 7! + 203e*" — —t —2t 3t _ 1 . I y=cje +c2e + c3e 9. y = c, cos (} In x) +c) sin ( In x) z= S5cejet + ce 72! + c3e3" M1. y=cyx 7 + cox * Inx 21. x = e733 — te73*3 — _ 4 - 3143 ~31+3 — \-1/2 1 (1 y=-e + 2te 13. y=x [c, cos( V31n x) + Cp sin(! V31n x)| 23. mx" =0 me . 15. y=er+c cos(V2 In x) +; sin(V2In x) my le _ X= Cj C2 17. y=c, + cox + €3x7 + eax? y=—tgh+ottcy 19. y=c, + x + Ex? Inx 21. y = cyx + cox Inx + x(In x)? — nyc _ qe eet ox Inx EXERCISES 4.9 (PAGE 177) ow «025. y= 2-2x? 27. y = cos(In x) + 2 sin(In x) WW 3. y = In|cos(c, — x)| + c e 29. y= $ —Inx+ ix 31. y =cyx 19 + cnx? 1 1 <x ' 5. y= slnicyx + 1] — —x + cy a 33. y= ex) + cox 8 + 5x" Ci C . 1 _ e =. 35. y = 27 [c, cos(3 Inx) + cy sinB Inn] + 44+ 3x ay —aysxtoe = 9(—yl?2 — 6(— 1/2 In $ 37. y = 2(—x) 5(—x)''* In(—x), x < 0 9, y =tan(} 2-43), -la<x<ia 4 1 5 I. y= --V1- cir +o cy a = EXERCISES 4.8 (PAGE 172) 3 yoltxtivtivgive ig... 1. x =cye' + cote’ Lu _— 1 2.3 14 7 te ce y = (cy — cade! + cote! 15. y=14+x-—5X 4+ 5x - 4x + ox + a 3. x =c,cost+cosint+tt+1 17. y=-V1-x 2 y=c,sint—cocost+t— 1 Z 5. x = $c, sint + $c, cost — 2c3sin V6t — 2c, cos V6t a y=c,sint+c cost + c3;sin V6t + cy cos V6t O | 2 3 4 CHAPTER 4 IN REVIEW (PAGE 178) a 7. x = ce + coe-*! + c38in 2t + cy, cos 2t + te! 1. y=0 Lu 5 y = ce! + ce! — c3sin 2t — cy cos 2t — tet 3. false Lo ” 5. (~~, 0); (0, %) 7 9. x =c, — c)cost + c; sint + 75e*' Te y = cye*® + ce + cyxe* + cge® + c5xe* + ce’: o y=c, + esint +c; cost — A es y= C18 + cox + 03x In x + cox + e5xIn x + cox (In x)” O = ¢ elt v3) a-v3) hi 11. x = ce’ + coe"? cos $ V3t + c3e7"? sin} V3t y= ce "+ ee ‘ n _ _ td y = (-3e, — 4 V3c3)e"”? cos} V3r HW. y=c) + ce * + c3xe™* . _ oo - 1 I s + (4 V30) — Sesle"? sin} V31 13. y=ce*? +e 3¥/2(c, cos} Vix + c3sin5 Vix) z 13. x= ce" + te 15. y = e*!2(c cost VI1x + c3 sind Vix) + 4x3 + 232 . 3 46 222 y= —icye" + cy + Set + iasX — 635 lo 1 4 1. x= ct ott e+ cet —4e 17. y =c, + cpe* + c3e** + ssinx — s;cosx + 3x y=(c,- cp +2) + (ep + Dt t+ yet —tP 19. y = e*(c,cos x + cy sin x) — e*cos x In|sec x + tan x| — yl 1/2 17. x = cye’ + Cpe"? sins V3t + c3e"? cos} V3t 21. y= cx + C2x t ( 1 13 ) 2 gin 3 23. y = cx? + cox? + x4 — x? Inx = ce t+ (-5¢e. —5 c3)e "* sins t . y ' wee 3 2 25. (a) y = c,cos wx + c)sin wx + Acos ax 1 _ 1 + (5 V3c) — as)e cos 5 V3t + Bsinax, w#ay; z=cet+ (—he, + 1V3¢,)e" sin} V3t y = c,cos wx + c)sin wx + Axcos wx + (-!1V3c, — }e;)e”? cos} V3t + Bxsinwx, w=a