Answer to Question #190487 in Differential Equations for Cecilia

Question

Complete solution of d^2y/dx^2 +2dy/dx +y=sin 2x

Answer 1

We have given the differential equation,

$\dfrac{d^2y}{dx^2} +2\dfrac{dy}{dx} +y=sin 2x$

$(D^2+2D+1)y = sin2x$

Auxiliary equation is:

$m^2+2m+1 = 0$

$(m+1)^2 = 0$

$m = -1,-1$

$CF = (C_1+C_2x)e^{-x}$

$PI = \dfrac{sin2x}{D^2+2D+1}$ $= sin2x\dfrac{1}{-4+2D+1}$

$= sin2x\dfrac{2D-3}{4D^2-9} = -\dfrac{1}{25}(2D-3)sin2x = -\dfrac{1}{25}(4cos2x-3sin2x)$

Hence the solution of given differential equation is :

$y = (C_1+C_2x)e^{-x}-\dfrac{1}{25}(4cos2x-3sin2x)$