 Endpoint obstruction in a homogeneous differential equation

For $x>0$ on a domain where the tangent is finite, put $v=y/x$. The homogeneous first-order differential equation becomes $x{v}^{\prime }=-\tan v$. Direct differentiation gives $(x\sin v{)}^{\prime }=0$, hence $x\sin (y/x)=C$, without dividing by $\sin v$ and losing constant-ratio solutions. If $C=x_0>0$ and $y/x\to \pi /2$ as $x\downarrow x_0$, two branches are $y=x\arcsin (x_0/x)$ and $y=x[\pi -\arcsin (x_0/x)]$, defined for $x>x_0$. They approach the same finite endpoint but have unbounded derivatives. The original tangent is undefined at the endpoint, so neither is a classical solution through that point. An implicit relation extending continuously to an initial point does not by itself solve an initial value problem there.