Question

Solve for the general solution using method of undetermined coefficients D4-1y=e-x

Answer 1

Find solution of the associated homogeneous equation:

$$(D^4-1)y=0$$

Characteristic (auxiliary) equation is

$$r^4-1=0$$

$$(r-1)(r+1)(r-i)(r+i)=0$$

$$r_1=-1, r_2=1, r_3=-i, r_4=i$$

The general solution of the associated homogeneous equation is

$$y_h=c_1e^x+c_2e^{-x}+c_3\cos x+c_4 \sin x$$

Find the particular solution of the non homogeneous differential equation

$$y_p=Axe^{-x}$$ $$y_p'=Ae^{-x}-Axe^{-x}$$

$$y_p''=-2Ae^{-x}+Axe^{-x}$$

$$y_p'''=3Ae^{-x}-Axe^{-x}$$

$$y_p''''=-4Ae^{-x}+Axe^{-x}$$

Substitute

$$-4Ae^{-x}+Axe^{-x}-Axe^{-x}=e^{-x}$$

$$A=-\dfrac{1}{4}$$

The particular solution of the non homogeneous differential equation is

$$y_p=-\dfrac{1}{4}xe^{-x}$$

The general solution of the non homogeneous equation is

$$y=c_1e^x+c_2e^{-x}+c_3\cos x+c_4 \sin x-\dfrac{1}{4}xe^{-x}$$