Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 6 / b / Solution

For the forcing $8e^{-2x}$, the method of undetermined coefficients gives the particular term $2e^{-2x}$. The term $e^{-x}$ is resonant with a homogeneous solution, so use $ax{e}^{-x}$ for its forcing; substitution gives $a=1$.

Thus $y=C_1{e}^{2x}+C_2{e}^{-x}+2e^{-2x}+x{e}^{-x}$. The two initial conditions reduce to $C_1+C_2=3$ and $2C_1-C_2=0$, so $C_1=1$, $C_2=2$. The solution of the initial value problem isThe extra factor $x$ in the resonant particular solution is essential.