Past exam of the mathematical methods I course of the University of Oxford / 2013 resit / 7 / Solution

For this homogeneous tangent differential equation, set $u=y/x$ on an interval not containing $x=0$. Since $y^{\prime }=u+x{u}^{\prime }$, cancellation gives $x{u}^{\prime }=2\tan u$. Away from zeros of $\tan u$, separation and integration giveRestoring a signed constant yields the compact implicit solutionA complete real branch description isHere $C$ is real and the interval must satisfy $x\ne 0$ and $|C{x}^2|<1$. The strict inequality excludes points with $\cos (y/x)=0$, where the original tangent function is undefined. Separating divided by $\tan u$, so its zeros must be checked separately: $u=k\pi$ gives $y=k\pi x$. These are exactly the $C=0$ branches already included. The branch integer is fixed on each connected solution interval; it cannot be changed arbitrarily at a regular point.