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

Factor the differential operator as $(D-a)(D-b)$, where $D=d/dx$. Its homogeneous solutions are $e^{ax},e^{bx}$, while a nonresonant particular solution is $e^{cx}/[(c-a)(c-b)]$. Imposing the two zero initial conditions gives the exponential forcing response with zero initial dataOne direct derivation avoids a large simultaneous system. Set $v=y^{\prime }-by$, so $v^{\prime }-av=e^{cx}$ and $v(0)=0$. The integrating factor gives $v=(e^{cx}-e^{ax})/(c-a)$; solving $y^{\prime }-by=v$ with $y(0)=0$ produces the displayed expression. This also verifies the zero derivative at the origin, since $v(0)=y(0)=0$.

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.