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

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