Please answer question 15 a and b with full explanations and also show all the work, there is a answer key is provided

Solutions to Math 34 Review Sheet 1. y' = y/(2(x + 1)) 2. (a) non-linear, first order, separable; (b) non-linear, first order, exact; (c) linear, first order, separable; (d) non-linear, first order, homogeneous of degree 4; (e) non-linear, first order. 3. (a) 2y? Iny - y? = 4re - 4ex - 1 (b) 2y? + x2 =: 916 (cy = - 320(22 +4)-4 4. (a) Iny? = x2 + y? + c2 (b) y? = 2x + C2 (c) 3 + y3 = 02 5. (a) fs(x) = 4f1(x) - 3/2(x), so linearly dependent; (b) Suppose c142 + c2x x] = 0; then from x = 1 we get c1 + c2 = 0 and from I = -1 we get c1 - C2 = 0; hence c1 = c2 = 0 and f1 and f2 are linearly independent; (c) W = -30esz * 0, so linearly independent. 6. (a) y2 = sinha; (b) y2 = 1; (c) 32 = e3z. 7. (a) y = cle2z + cze"; (b) y = ciesz + calebz; (c) y = cie(-3 + v29)/2 + cze(-3 - V29)/2; (d) y = e2z (c1 cos I + c2 sin x); (e) y = v3 cost +- sint. 8. (a) y = cie-3 + cze2z - x/3- 1/18; (b) y = c1 cos 2x + c2 sin 2x - (3/4)x.cos 2x. 9. (a) y = cle *+cze 2 +(e " te-2x) In(1+e7); (b)y = cir+c2x Inx+(2/3)x(Inx)3 10. (a) r = clet +cate, y = (c1-c2)et +czter; (b).x = C1-62 cos t+ c3 sin t+ (17/15) e3t, y = c1 + c2 sint+ c3 cost- (4/15)33t; (c) x = cjet + cze_t/2 sin(v/3t/2) + cae-t/2 cos(V3t/2), y = clet + (-c2/2 - V3c3/2)e-t/2 sin(V3t/2) + (v3c2/2 - c3/2)e-t/2 cos(V3t/2), z = ciet + (-c2/2 + v3c3/2)e-t/2 sin(V3t/2) + (-V3c2/2 - c3/2)e-t/2 cos(v3t/2). 11. (a) -3e-s/s - 2e-s/s2 + 2/s2 + 1/s; (b) 1/(s - 4)2; (c) 252 49 + $2 + 1) 12. (a) 2 cos 3t -- 2 sin 3t; (b) (3e-3t + et)/4; (c) ;U(t - [) sin 3(t - 7). 13. 2ks/(82 + k?)2. 14. (s/ (s' + 4) )e -"s 15. (a) y = the2t /20; (b)y = cos 2t - { sin 2(t - 27)U(t - 27) + $ sin(t - 27) U(t - 2x). 16. (a) yi(x) = co[1 + 23/(3 . 2) + 26/(6 . 5 . 3 . 2) + 29/(9 . 8 . 6 . 5 . 3 .2) + . .. ], y2(T) = ci[r + 24/ (4 . 3) + x7/(7 .6 :4 .3) + x10/(10 .9 .7 . 6 . 4 .3) +. . .]; (b) yl(x) = coll - 23 /3!+42x6 /6!-72 . 42x9/9! + . . .], yz(x) = ci[x -22x4/4!+52 .22/7!-82.52.22x10 /10!...) 17. y = cli=1 + C2x-8 18. If n, m are odd then " sin na sin ma = (1/2) for cosin-m)x-cos(n+m)r dr = (1/2)[(1/(n - m)) sin(n - m)r - (1/(n + m)) sin(n + m)xjo Jack/2 =: 0 since m +n and m - n are even. 19. (a) f(x) = + En=1 1-(-1) COS nix - sin nix n272 (-1 )n+1 (b) f (z) = 1+ En=1 { (-1)" - 1 - COS nix + - sin nix 1272 12/97:GW8. Use the method of undetermined coefficients to solve (a) y" + y' - by = 2x (b) y" + 4y = 3 sin 2x 9. Use the method of variation of parameters to solve the following: (a) y" + 3y' + 2y = 1+ ex 1 (b) chy" - ry' t y = 4x Inr given that y = x and y = In a form a fundamental set of solutions to ry" - ry' ty = 0. 10. Solve the following systems of linear differential equations: = 2x - y (a) dt dy ( b ) Dr + Day = e3t Dr = y (D +1)r + (D-1)y = 4e3t. . ( c) I Dy = z Dzz x 11. Determine the Laplace transform of each of the following functions: ( a) f (t ) = 2t + 1, Ost