Question

There are 3 problems in the first part of this homework assignment. Consider the following second order nonhomogeneous linear dif- ferential equations (1) y" + 100y = 36 cos(8x) + 72 sin(8x), (2) y" – 4y' + 3y = (2 – 12x)e" + (60 + 40x)e37, (3) Y" – 2ay' +a²y = 60xeQ2 + 60x’ear, where a + ( is a real nonozero constant. Use the method of unde- termined coefficients to find a particular solution for each equation. Then solve each equation for real general solution.

please do problem No. 3!

Answer 1

ra 9 d 1 20 Y" - 2ay' +22y = 6oo ex+60 8² cdx 6 (D² - 2a D to?) y = 60xeaz + 60x²e ax { 2 - 0 = (Gox + 60x2) eax Aune liary exaution le m² - 2amta2 = 0 (m-a) 2 = 0 m=a,& the complementary solution is y = (6 + G₂ x) e xx for the particular solution 18 Jp = (AX+BX+Clear Yo = (PAX +8) ***+ *e ** ( AX +8&+C) yp = PALM4 CAX+ Bjxe ax + one ax(Ax®+cerc) et dean (PARAB) pet equation or 2²0** ( AKCE BACH) + adeo/2A%+B) + 2 Aed - Od(2A2+ B)e«X_2d2farl ALP+BX+C) + PAX +BX+C)60x ***+bonierte 2 A0%= 60 x eadt60x² exa A= 30x + Box² not solve 20

A second grape for the particular solution is yp = 2 (Ax2+BX+C) ease and the final Yp le Jp = (10x3 + 5x+) e da the general solution y = Je + Zp y = (6 + cp ) 08%+ 10 x ² ( x + 58+ (2x y = ((, +G* + 10x3 +507) e a