Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 4 / b / Solution

The characteristic equation of a constant-coefficient differential equation here is $r^2-3r+2=(r-1)(r-2)=0$. Its homogeneous solution is $C_1{e}^x+C_2{e}^{2x}$. Because the forcing $e^x$ is already a homogeneous mode, the method of undetermined coefficients requires $y_p=Ax{e}^x$ rather than $A{e}^x$.

Here $y_p^{\prime }=A(x+1)e^x$ and $y_p^{\prime \prime }=A(x+2)e^x$, so $y_p^{\prime \prime }-3y_p^{\prime }+2y_p=-A{e}^x$. Thus $A=-1$, and the general solution is