Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 15 / a

The characteristic equation of the homogeneous linear differential equation isThe real general solution is $y=e^{2x}(A\cos 3x+B\sin 3x)$. The first boundary value gives $A=\pi$. At $x=-\pi /2$, $\cos (-3\pi /2)=0$ and $\sin (-3\pi /2)=1$, so $1=e^{-\pi }B$. Hence

The characteristic equation has a repeated root $-1$, suggesting the substitution $y=e^{-x}v$. By the product rule,so $y^{\prime \prime }+2y^{\prime }+y=e^{-x}{v}^{\prime \prime }$. The inhomogeneous linear differential equation reduces to $v^{\prime \prime }=2x$. Two integrations giveThereforeThe two arbitrary constants account for the repeated-root homogeneous solution.