Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 2 / 7 / b

Write $y(\theta )=Y(x)$ with $x=\cos \theta$. By the chain rule,The terms involving $Y_x$ cancel in the original ordinary differential equation, leavingOn each interval where $\sin \theta \ne 0$, the transformed linear ordinary differential equation is $Y_{xx}+2Y=0$. Its characteristic equation has roots $\pm i\sqrt{2}$, yieldingDirect substitution shows these expressions also extend smoothly through $\theta =k\pi$ and satisfy the original equation there. The two constants describe the general solution on any interval where the coordinate substitution is invertible.