Exercícios Selecionados Livro Calc2 Pt3-2021 2

148 e CHAPTER 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS REMARKS (1) In Problems 27-36 in Exercises 4.4 you are asked to solve initial-value problems, and in Problems 37-40 you are asked to solve boundary-value problems. As illustrated in Example 8, be sure to apply the initial conditions or the boundary conditions to the general solution y = y, + yp. Students often make the mistake of applying these conditions only to the complementary function y, because it is that part of the solution that contains the constants Cily Cy 0 0 0 9 G0 (ii) From the “Form Rule for Case I” on page 145 of this section you see why the method of undetermined coefficients is not well suited to nonhomogeneous linear DEs when the input function g(x) is something other than one of the four basic types highlighted in color on page 141. For example, if P(x) is a polyno- mial, then continued differentiation of P(x)e™ sin Bx will generate an indepen- dent set containing only a finite number of functions —all of the same type, namely, a polynomial times e°* sin Bx or a polynomial times e°* cos Bx. On the other hand, repeated differentiation of input functions such as g(x) = In x or g(x) = tan” |x generates an independent set containing an infinite number of functions: derivatives of | eet 2 erivatives of In x: ia yer ar ae derivatives of tan! 1 2) 6x" erivatives of tan” 'x: Ss oe I+xr 1t+xy d+x) EX E RC | S E S 4 . 4 Answers to selected odd-numbered problems begin on page ANS-5. In Problem{1—26)holve the given differential equation by 16. y" — 5y’ = 2x9 — 4x? -x +6 undetermined coefficients. 17. y" —2y' + Sy = e* cos 2x 1. y" + 3y' + 2y=6 18. y" — 2y' + 2y = e**(cos x — 3 sin x) 2. 4y" + 9y = 15 19, y" + 2y’ + y = sin x + 3 cos 2x 3. y"— 10y" + 25y = 30x + 3 20. y" + 2y' — 24y = 16 — (x + 2ye** 4. y"+ y' — 6y = 2x 21. y” — 6y” =3 —cosx 1 22. y” — 2y"” — 4y’ + 8y = 6xe** 5. -y"+y +y=x?-2x y y y y 4 23. y” — 3y" + 3y' —~y =x — 4e* 6. y" _ 8y' 4 20y _ 100x2 — 26xe* 24. ym _ y" _ Ay’ + 4y =5—er4+ ek 7. y" + 3y = —48x7e3* 25. y+ 2y"+y=(x- 1)? 8. 4y” — 4y’ — 3y = cos 2x 26. y — y" = 4x + 2xe 9, y" _ y' _ 3 In Problem(27-39 solve the given initial-value problem. 10. y" + 2y’ =2x+5-e* ” , 1 ” — a — 1 ! a _— 11. y"—y taya3 te" 27, y" + 4y = —2, 2) =3y(2) =2 12. y" — 16y = 2e* 28. 2y" + 3y’ — 2y = 14x7-4x-11, y(0) =0, y'(0) =0 13. y" + 4y = 3 sin 2x 29. 5y" + y' = —6x, y(0) = 0, y’(0) = —10 14. y" — 4y = (x? — 3) sin 2x 30. y" + 4y' + 4y=834+xe, yO) =2,y'(0)=5 15. y" + y = 2x sinx 31. y" + 4y’ + 5y = 35e*, y(0) = -3, y'(0) = 1 4.4. UNDETERMINED COEFFICIENTS—SUPERPOSITION APPROACH e 149 32. y’—y=coshx, y(0) =2,y'(0) = 12 45. Without solving, match a solution curve of y” + y = f(x) ax shown in the figure with one of the following functions: 33. We + wx = Fysin at, x(0) = 0,x'(0) =0 () f(x) = 1, (ii) f(x) =e”, f (iii) f(x) = e*, (iv) f(x) = sin 2x, ad — ot . Gc 34, + wx = Fycos yt, x(0) = 0,x'(0) = 0 (v) Fe) = eb sinx, (vi) f(x) = sin x. dt Briefly discuss your reasoning. 35. ym _ 2y" 4 y' =2—24e% + 40e™, y(0) = , y'0) = 3.9") = -3 (a) 7 36. y"” + 8y =2x-—5+8e**, y(0) = —5, yO) = 3, y"(0) = ~4 In Problem(37—4)solve the given boundary-value problem. 37. y"t+y=x°+1, y)=5,y() =0 38. y” — 2y' + 2y=2x-2, y(0)=0,y(7) =7 39. y” + 3y = 6x, y(0) = 0, y(1) + y’1) = 0 FIGURE 4.4.1 — Solution curve 40. y" + 3y = 6x, y(0) + y'(0) = 0, y1) = 0 y In Problems 41 and 42 solve the given initial-value problem (b) in which the input function g(x) is discontinuous. [Hint: Solve each problem on two intervals, and then find a solution so that y and y’ are continuous at x = 7 /2 (Problem 41) and at x = am (Problem 42).] y” + 4y = g(x), yO) = 1, yO) = 2, where x sinx, OSxS 7/2 FIGURE 4.4.2 Solution curve 80) = 0, x > 7/2 (c) » y” — 2y'’ + 10y = g(x), y(O) = 0, y’(0) = 0, where x (x) 20, O=Sx=T7 x) = 6 0, x«>T FIGURE 4.4.3 — Solution curve Discussion Problems 43. Consider the differential equation ay” + by’ + cy = e™, (d) y where a, b, c, and k are constants. The auxiliary equation of the associated homogeneous equation is am? + bm+c=0. (a) If k is not a root of the auxiliary equation, show x that we can find a particular solution of the form yp = Ae*, where A = 1/(ak? + bk + c). FIGURE 4.4.4 Solution curve (b) If k is a root of the auxiliary equation of multiplicity one, show that we can find a particular solution of the form y, = Axe*, where A = 1/(2ak + b). Computer Lab Assignments Explain how we know that ke ~b/ (2a). . In Problems 46 and 47 find a particular solution of the given (c) If kis a root of the auxiliary equation of multiplicity differential equation. Use a CAS as an aid in carrying out two, show that we can find a particular solution of the differentiations, simplifications, and algebra. form y = Ax?e**, where A = 1/(2a). 46. y" —4y' + =(2 2 2x 2 44. Discuss how the method of this section can be used 6. y y By = (2x pve cos oe . . . ” : + (10x* — x — 1)e** sin 2x to find a particular solution of y” + y = sin x cos 2x. Carry out your idea. 47, yO + 2y" + y =2cosx — 3xsinx 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 _ =) e738 4 (1 +4 =) eV3e: 41. y=c, +Ox+coe* + = x4 — Sx + 8x? 2 V3 2 V3 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’ — gre’ + gxe’ — 5 L. y=cye® + pe 2* +3 53. y = e*(c,cos 2x + cysin 2x) + fe sin x ~yrel 2 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* + cyxe* — x — 3 — Exe 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 wo o 9. y= ce +t oe ¢ fer] rs nae O _ : else -y=ce coe ** + gf e?* In|x| — e dt}, uw 35. y = 11 — lle* + 9xe* + 2x — 12x e + 5e xy ot a 37. y = 6 cos x — 6(cot 1) sinx + x7 — 1 Xo > O o 39 —4 sin V3x 4 11. y= ce + ce 2% + ( + e7?*) In(1 + e*) O . = _ —2x —x 72x go x y sin V3 + V3 cos V3 x 13. y=cje “+ c2e e ‘sine an 5 1 15. y=cje'+ ote! + iPe! Int — +Pet co cos 2x + ?sin2x + ;sinx, OSx< 7/2 . ' : — 41. y=}, dx + Sein? > a/2 17. y = cye*sinx + c,e* cos x + 4xe* sin x = 5 cos 2x + 2 sin 2x, x> 7 n ° ° + 4e*cos x In| cos x| Zz 19. y= 1 oad 4 3 pri2 4 12 oxl2 _ 1, x2 . 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? : yD 5) + ay =x—6 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)