Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 14B (Further Complex Methods)

For the Laplace-type contour ansatz, differentiating under the integral and putting $P(t)=t^2-t-2$ gives the differential-equation residualIntegration by parts turns this into $[e^{zt}Pf{]}_{\partial \gamma }+{\int }_{\gamma }{e}^{zt}[(2-t)f-(Pf{)}^{\prime }]dt$. Thus it vanishes whenever the endpoint contribution vanishes and $(Pf{)}^{\prime }=(2-t)f$. The latter condition is $P{f}^{\prime }=3(1-t)f$, solved byup to a constant multiplier. Closed contours avoiding the poles make the endpoint contribution zero.

Choose separate positively oriented small circles around 2 and $-1$. The residue theorem gives, after division by $2\pi i$,ThereforeThe Wronskian of the two basis functions is $-9z{e}^z$, nonzero for $z\ne 0$. They span the two-dimensional solution space on every regular domain, and both extend entire across the apparent singular point at zero. Their initial values satisfy $w(0)=A+B$ and $w^{\prime }(0)=2(A+B)$. Thus every solution has $w^{\prime }(0)=2w(0)$, and the requested combination $w(0)=0$, $w^{\prime }(0)\ne 0$ is impossible. No additional logarithmic solution is lost: the two independent functions already span the solution space away from zero.