Question

Find the general solutions to each of the following

1) y''-2y'-3y=2e^4x

Answer 1

$$y''-2y'-3y=2e^{4x}$$

Related homogeneous differential equation

$$y''-2y'-3y=0$$

The roots of the characteristic equation are

$$r^2-2r-3=0$$

$$(r+1)(r-3)=0$$

$$r_1=-1,r_2=3$$

The general solution of the homogeneous differential equation is

$$y_h=c_1e^{-x}+c_2e^{3x}$$

Find the particular solution of the nonhomogeneous differential equation

$$y_p=A e^{4x}$$

$$y_p'=4Ae^{4x}$$

$$y_p''=16Ae^{4x}$$

Substitute

$$16Ae^{4x}-8Ae^{4x}-3Ae^{4x}=2e^{4x}$$

$$A=\dfrac{2}{5}$$

$$y_p=\dfrac{2}{5}e^{4x}$$

The general solution of the nonhomogeneous differential equation is

$$y=y_h+y_p$$

$$y=c_1e^{-x}+c_2e^{3x}+\dfrac{2}{5} e^{4x}$$