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

The characteristic equation $(r-3)(r-4)=0$ gives complementary terms $C{e}^{3x}+D{e}^{4x}$. A linear particular solution is $12x+7$, since its substitution gives $144x$. The initial conditions give $C+D=-7$ and $3C+4D=-12$, hence

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

The repeated characteristic root is $-1$, matching the exponential forcing. Put $y=e^{-x}w$; then $y^{\prime \prime }+2y^{\prime }+y=e^{-x}{w}^{\prime \prime }$, so $w^{\prime \prime }=8$. The zero initial conditions give $w(0)=w^{\prime }(0)=0$, henceThe extra quadratic factor is the resonance associated with the double root.