Solve the initial value problem

$y^{\text{'}\text{'}\text{'}}-2y^{\text{'}\text{'}}+y^{\text{'}}=1+x{e}^x,y(0)=y^{\text{'}}(0)=0,y^{\text{'}\text{'}}(0)=1.$

Find the general solution $of$ the corresponding homogeneous

differential equation.

Find the general solution $of$ the inhomogeneous differential equation

using the method $of$ Undetermined Coefficients. Explain why you set

$up$ the inhomogeneous equation $in$ the way you did. Then find the

solution $of$ the given initial value problem.