Answer to Question #207804 in Differential Equations for abigail

Question

y''+y=senx

Answer 1

$y''+y=\sin x$

Write the related homogeneous or complementary equation:

$y''+y=0$

The general solution of a nonhomogeneous equation is the sum of the general solution $y_h(x)$ of the related homogeneous equation and a particular solution $y_p(x)$ of the nonhomogeneous equation:

$y(x)=y_h(x)+y_p(x)$

Consider a homogeneous equation 

$y''+y=0$

Write the characteristic (auxiliary) equation:

$r_1=i, r_2=-i$

The general solution of the homogeneous equation is

$y_h(x)=C_1\sin x+C_2\cos x$

Let

$y_p=x(A\sin x+B\cos x)$

Then

$y_p'=A\sin x+B\cos x+x(A\cos x-B\sin x)$

$y_p''=A\cos x-B\sin x+A\cos x-B\sin x$

$+x(-A\sin x-B\cos x)$

Substitute

$A\cos x-B\sin x+A\cos x-B\sin x$

$+x(-A\sin x-B\cos x)+x(A\sin x+B\cos x)$

$=\sin x$

$2A\cos x-2B\sin x=\sin x$

$A=0, B=-\dfrac{1}{2}$

The general solution of a second order homogeneous differential equation be

$y(x)=C_1\sin x+C_2\cos x-\dfrac{1}{2}x\cos x$