Answer to Question #251560 in Differential Equations for Shagor

Question

Solve by the method of underminded coefficient using superposition apporoch

(D^2 +2D+2)y=2e^-x

Answer 1

Homogeneous differential equation

$$(D^2 +2D+2)y=0$$

Corresponding (auxiliary) equation

$$r^2+2r+2=0$$

$$(r+1)^2=-1$$

$$r=-1\pm i$$

The general solution of the homogeneous differential equation is

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

Find the particular solution of the nonomogeneous differential equation

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

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

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

Substitute

$$Ae^{-x}-2Ae^{-x}+2Ae^{-x}=2e^{-x}$$

$$A=2$$

$$y_p=2e^{-x}$$

The general solution of the given nonhomogeneous differential equation is

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