Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 4

The specified range ensures $\cos y\ne 0$, so the separable differential equation can be written ${\sec }^{2}ydy=\sin xdx$. Integrating gives $\tan y=C-\cos x$. The range selects the principal inverse tangent, henceEvery finite $C$ produces a solution within the required range. Constant solutions at the excluded values $y=\pm \pi /2$ are not part of this family or of the question's domain.

The general solution of this first-order linear ordinary differential equation is $y=C{e}^{3x}$. Its initial condition gives $C=3$, soDifferentiation verifies $y^{\prime }=3y$ and substitution at zero gives the required value.