Question

Find the general solution of the following differential equations using method of undetermined coefficients: (ii) y''−2y'+y =20e^xsinx,

Answer 1

Corresponding homogeneous differential equation

$$y''−2y'+y =0$$

Characteristic (auxiliary) equation

$$r^2-2r+1=0$$

$$r_1=r_2=1$$

The general solution of the homogeneous differential equation is

$$y_h=c_1e^x+c_2xe^x$$

Find the particular solution of the non homogeneous differential equation

$$y_p=Ae^{x}\cos x+Be^x\sin x$$

$$y_p'=Ae^x\cos x-Ae^x\sin x+Be^x\sin x+Be^x\cos x$$

$$y_p''=Ae^{x}\cos x-2Ae^x\sin x-Ae^{x}\cos x$$

$$+Be^x\sin x+2Be^x\cos x-Be^x\sin x$$

Substitute

$$-2Ae^x\sin x+2Be^x\cos x-2Ae^x\cos x$$

$$+2Ae^x\sin x-2Be^x\sin x-2Be^x\cos x$$

$$+Ae^{x}\cos x+Be^x\sin x=20e^x\sin x$$

$$A=0$$

$$B=-20$$

The general solution of the non homogeneous differential equation is

$$y=c_1e^x+c_2xe^x-20e^x\sin x$$