Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 2 / 13 / ii / Solution

The complementary characteristic roots are $-1,-2$. Substituting $A\sin x+B\cos x$ into the differential operator gives $(A-3B)\sin x+(3A+B)\cos x$, so a particular solution is $\sin x-3\cos x$. The initial conditions then require $C+D=3$ and $C+2D=1$, giving