Past exam of the mathematical methods I course of the University of Oxford / 2019 resit / 7 / Solution

Work on an interval with $x\ne 0$ and $\cos (y/x)\ne 0$, as required by the original equation. Set $v=y/x$, so $y^{\prime }=v+x{v}^{\prime }$, and obtain $x{v}^{\prime }=3\tan v$. Where $\sin v\ne 0$, separation of variables gives $\cot vdv=3dx/x$ and thereforeThe solutions lost by division by $\tan v$ are $v=k\pi$, and they are included by $C=0$. On a fixed local branch one can writeThe integer $k$ is fixed along a connected regular branch. Implicit differentiation recovers the original equation whenever $x\cos (y/x)\ne 0$. The boundary $|C{x}^3|=1$ is excluded because the original tangent function has a pole there.