Past exam of the year 1 of the computer science course of the University of Oxford / 2005 / Calculus, linear algebra, discrete mathematics, logic and proof, and models of computation / 3 / a / i

The homogeneous linear differential equation has solutions $\cos x$ and $\sin x$. Since the forcing is already a homogeneous solution, seek a particular solution proportional to $x\cos x$. Direct differentiation givesThus the general solution of the linear ordinary differential equation isThe first initial condition gives $A=0$. Alsoso the second initial condition gives $B-\frac{1}{2}=-\frac{1}{2}$, hence $B=0$. Therefore