= Second derivative identity for a degree-one homogeneous function
{title2=$x^2h_{xx}=y^2h_{yy}$}
For a twice <continuously differentiable> <homogeneous function> $h(x,y)$ of degree one, <Euler theorem for homogeneous functions> gives $xh_x+yh_y=h$. Differentiation first with respect to $x$ and then, separately, with respect to $y$ gives
$$
xh_{xx}+yh_{xy}=0,\qquad xh_{xy}+yh_{yy}=0.
$$
Multiplying by $x$ and $y$, respectively, shows that $x^2h_{xx}=y^2h_{yy}=-xyh_{xy}$. This identity can verify a second-order <partial differential equation> without expanding a complicated homogeneous expression twice.
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