Past exam of the mathematical methods I course of the University of Oxford / 2013 resit / 12 / b

Again put $u=y/x$, requiring $x\ne 0$. Substituting $y^{\prime }=u+x{u}^{\prime }$ gives $x{u}^{\prime }=1+u^2$. For real $u$ the factor $1+u^2$ never vanishes, so separation loses no constant real solution:Each real solution is defined on a connected interval avoiding zero and every pole of the tangent function. Changing the constant by an integer multiple of $\pi$ gives the same branch. If complex-valued solutions are also allowed, the separated-out constant values $u=\pm i$ supply the additional solutions $y=\pm ix$; these do not occur in the real-valued family.