Question

(27 points) Find the general solution of the associated homogeneous equation for each nonhomogeneous differential equation below. Then determine the form of a particular solution ур of the nonhomogeneous equation. Do not solve for the undetermined coefficients in yp (a) (10 points) y" – 9y' – 22 y = 5xe -2x (b) (10 points) y" – 4y' + 29 y = 8x sin 3x

Answer 1

solution : (a) Given is: Differential Equation y" - qy' - 224 sxe 2x Corresponding homogenous equeation is (2) 975 y"_ qy' - 224 Its Quriliary equcation m²- am - 22 is 0 m? lim+ am 22=0 mim-il) + 2 (mall) =0 fon (m-ll) (m +2) 20 ma 11, -2 we + e -2x where Now have real and distinct roots of the Quwiliony equation .. complementary Soledwin y clx) will be Yux) C, e C, and C2 are arbitrary constants let Yp denotes the particular Then by the method of undetermined Coefficients the particular solution of the form x 1 a, e aze (2) solution of will be 27 Yp (x) = + 2x 27 where a; e +.Que is multiplied by X to account for e-za in the

.. 2x A * Complementary solution present RHS on of a also. This form of particular solection Spixs is 23 24 Iply) xl a e . a, e w ) and general homogenous associated solution for equation is YUX) TIX Cie - 2x +(2 e genous (6) Giver Differential equation is y" - 44' +297 = 8x sinun gts associated homo differential equation y"- uy' +294 -(2) 9ts auxiliary equation is m²- um +29=0 by Quadratic formula (-4) +-4-4114291 Nou 4 trio- 4%29 2 4 sury-) 4 + 25-25 22

Lar 4.-.. m (2 + 5i) 2 + si .. we have Complex roats :. the complementary solution 4c1x) win be e27 Now 9: (X) = (a cas 5x + (2 Sin 5x) let Yolx) denotes the particular of (1) 3 3 solution the uncleterminant Then by Coefficients method of Splas will he of 3 form 3 Uplo) 3 a, cas(3x)+ ax cos(3x) + Az Sin (34) + aun sin(3) Thus the associated general solution of equation e2" (cicos 5x + ca sin 5x) homo genous Yc(X) = is and the form of particular solution Yola) will be 4p(1) - a, cos (32) + OX COS (3X) + a₂ sin(3x) + aux sin (3x)

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