Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 10 / b / Solution

Now $c=b$, so the forcing coincides with a homogeneous exponential and requires a resonant exponential particular solution. With $v=y^{\prime }-by$ as above, $v^{\prime }-av=e^{bx}$ and $v(0)=0$ give $v=(e^{ax}-e^{bx})/(a-b)$. ThereforeIntegrating from zero yieldsIts numerator starts at order $x^2$, ensuring both initial conditions. It is also the finite coalescing-rate limit $c\to b$ of the nonresonant exponential forcing response with zero initial data.