Past exam of the mathematical methods I course of the University of Oxford / 2021 resit / 10 / d / Solution

The homogeneous linear differential equation has roots $(-1\pm i\sqrt{3})/2$, so its complementary function decays:For a particular integral, set $f_p=B\cos 2t+C\sin 2t$. Substitution givesThe two coefficient equations are $-3B+2C=A$ and $-2B-3C=0$, so $B=-3A/13$ and $C=2A/13$. Consequently every solution obeysThe limiting behaviour is a periodic steady oscillation with amplitude $|A|/\sqrt{13}$, the long-time response of a forced harmonic oscillator. Unless $A=0$, there is no finite numerical limit: at $t=n\pi$ and $t=n\pi +\pi /2$, the two limiting values are $-3A/13$ and $3A/13$. For $A=0$, $f(t)\to 0$.