Answer to Question #302581 in Differential Equations for haru

Question

Find the general solution of the following differential equations using method of undetermined coefficients: (iv) y''−y =2sinx+2

Answer 1

Corresponding homogeneous differential equation

$$y''−y =0$$

Characteristic (auxiliary) equation

$$r^2-1=0$$

$$r_1=1,r_2=-1$$

The general solution of the homogeneous differential equation is

$$y_h=c_1e^x+c_2e^{-x}$$

Find the particular solution of the non homogeneous differential equation

$$y_p=A\cos x+B\sin x+C$$

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

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

Substitute

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

$$A=0, B=-1, C=-2$$

The general solution of the non homogeneous differential equation is

$$y=c_1e^x+c_2e^{-x}-\sin x-2$$