Past exam of the mathematical methods I course of the University of Oxford / 2016 / 9 / c / Solution

The characteristic equation of the homogeneous ordinary differential equation is $r^2-2r+5=0$, with roots $1\pm 2i$. Therefore $y_h=e^x(C\cos 2x+D\sin 2x)$. For the forcing term, try the particular solution $y_p=e^{-x}(ax+b)$. Substitution gives $8a=1$, $8b-4a=0$, so $a=1/8$, $b=1/16$. The two initial conditions give $C=-1/16$ and $D=0$. ConsequentlyBoth the value and the first derivative vanish at zero; direct differentiation verifies the inhomogeneous equation.