= Endpoint obstruction in a homogeneous differential equation
{title2=$y'=y/x-\tan(y/x)$}
For $x>0$ on a domain where the tangent is finite, put $v=y/x$. The <homogeneous first-order differential equation> becomes $xv'=-\tan v$. Direct differentiation gives $(x\sin v)'=0$, 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.
Back to article page
