Question

Determine the form of a particular solution for each differential equation. No need to solve for the general solution of the differential equation.

1. y'' − 3y' + 2y = xe^x + 1

Answer 1

Corresponding homogeneous differential equation

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

Characteristic (auxiliary) equation

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

$$r_1=1, r_2=2$$

Find a particular solution in form $y_p=x(Ax+B)e^x+C.$

$$y_p=Ax^2e^x+Bxe^x+C$$

$$y_p'=Ax^2e^x+2Axe^x+Bxe^x+Be^x$$

$$y_p''=Ax^2e^x+4Axe^x+2Ae^x+Bxe^x+2Be^x$$

$$Ax^2e^x+4Axe^x+2Ae^x+Bxe^x+2Be^x$$

$$-3Ax^2e^x-6Axe^x-3Bxe^x-3Be^x$$

$$+2Ax^2e^x+2Bxe^x+2C=xe^x+1$$

$$-2A=1$$

$$2A-B=0$$

$$2C=1$$

$$A=-1/2, B=-1, C=1/2$$

The particular solution is

$$y_p=-\dfrac{1}{2}x^2e^x-xe^x+\dfrac{1}{2}$$