Question

PROBLEM 37: Find the general solution to inhomogeneous ODE y" 3y 2y 4t using the method of undetermined coefficients with the guess yp = At + B PROBLEM 38: Solve the inhomogeneous ODE 13 cos(2t) y" 7y12y + using the method of undetermined coefficients PROBLEM 39: Find the general solution for y"4y4y exp(-2t) + using the method of undetermined coefficients

Answer 1

Que. 37 y + 3y' tay = 4t Auxiliary eqh D² +32 2 42 20 D -1, -2 Ya Ciert +Cl2 at rat ce(+1+9+430 - ): Ip. = 2 [++ ] = at +3 Gs - че ус + Ур yaciet te eet +at+3 due. 38 y" + 7y' +12y = 13 cool2t) Auxiliary equ A²+70+1220 0 2-3,-4 Yea Ce-3t +C₂e-4t yo 3 cosat put 2 – 22 D² +70 H12 -4. 13 cost = 13 (708) coset Yo +12-4 Co+8) 670-6) - 13(70-8) coast 49 D ² - 64 a 13(70-8) coat - 13(708) cost - 260 49(-4)-64 20 =:-((-zsinet) – 8 coszt) s to (-14 Sinet -8 coszt) Ipal (7sinat + 4 cost) « 9° 9% +Ур — y = ce-3t + C₂ e 4t I (tsinat + 4 cost) Oue 39 44 +4y=enzt. y'+44 +4y=e Auxiliary eqh, Q2 +40+450 A = -2,-2 Ye = (6, +Cat) e

D²+40+4 put D2-2. 24 e (-2) ² + 4 (-2) +4 MG put wan2. 2070 erat. Ypa t ezt GSyYc typ ya (@, + (at) e-zt + t² e 2t. ya (e,t Cat + t ) e

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