Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 13 / a

An integrating factor is $e^{3x}$. Multiplication gives $(e^{3x}y{)}^{\prime }=8e^{3x}$, so $y=8/3+C{e}^{-3x}$. The initial condition gives $C=4/3$. Therefore

First write the forcing as $\sin x\cos x$. An integrating factor for the linear ordinary differential equation is $e^{-\sin x}$, givingSet $u=\sin x$ inside the antiderivative. ThenThe initial value forces $C=1$, henceThe resulting expression is explicit and valid for all real $x$; the substitution does not impose a global monotonicity condition on $\sin x$ when verifying an antiderivative.