Question

Find the general solution of the equation d²y/dx²+4dy/dx+4y=4e^-2x

Answer 1

Related homogeneous differential equation

$$y''+4y'+4y=0$$

The roots of the characteristic equation are

$$r^2+4r+4=0$$

$$(r+2)^2=0$$

$$r_1=r_2=-2$$

The general solution of the homogeneous differential equation is

$$y_h=c_1e^{-2x}+c_2xe^{-2x}$$

Find the particular solution of the nonhomogeneous differential equation

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

$$y_p'=2Axe^{-2x}-2Ax^2e^{-2x}$$

$$y_p''=2Ae^{-2x}-8Axe^{-2x}+4Ax^2e^{-2x}$$

Substitute

$$2Ae^{-2x}-8Axe^{-2x}+4Ax^2e^{-2x}$$

$$+8Axe^{-2x}-8Ax^2e^{-2x}+4Ax^2 e^{-2x}=4e^{-2x}$$

$$A=2$$

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

The general solution of the nonhomogeneous differential equation is

$$y=y_h+y_p$$

$$y=c_1e^{-2x}+c_2xe^{-2x}+2x^2 e^{-2x}$$