Question

Undetermined Coefficients

4. y^ prime prime +2y^ prime +5y=1+e^ x

Answer 1

$$y''+2y'+5y=1+e^x$$

The characteristic (auxiliary) equation for homogeneous differential equation is

$$r^2+2r+5=0$$

$$(r+1)^2=-4$$

$$r=-1\pm 2i$$

The general solution of the homogeneous differential equation is

$$y_h=C_1e^{-x}\cos 2x+C_2e^{-x}\sin 2x$$

Find a particular solution of the nonhomogeneous differential equation. 

$$y_1=Ae^x+B$$

$$y_1'=Ae^x$$

$$y_1''=Ae^x$$

Substitute

$$Ae^x+2Ae^x+5Ae^x+5B=1+e^x$$

$$A=\dfrac{1}{8}, B=\dfrac{1}{5}$$

$$y_1=\dfrac{1}{8}e^x+\dfrac{1}{5}$$

The general solution of the given differential equation is

$$y_h=C_1e^{-x}\cos 2x+C_2e^{-x}\sin 2x+\dfrac{1}{8}e^x+\dfrac{1}{5}$$