Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 8 / Solution

The characteristic equation is $(r-1{)}^2=0$, so the repeated-root constant-coefficient differential equation gives the general solution $y=(A+Bx)e^x$. The first boundary condition gives $A=0$, and the second gives $B\pi e^{\pi }=\pi e^{\pi }$, hence $B=1$. Therefore