 Solution

The first-order chain rule, with $x,y$ regarded as functions of $u,v$, isApply the product rule when differentiating the first identity in $v$, and apply the chain rule to $f_x,f_y$. Equality of mixed partial derivatives for a $C^2$ function givesThus $\boxed{H=x_{uv},K=y_{uv}}$. These terms record the second derivatives of the coordinate transformation, not of $f$.

For the given transformation,The mixed-derivative coefficient is zero, while $x_u{x}_v=x$, $y_u{y}_v=-y^2/x$, and $1/v^2=y/x$. The required partial differential equation is therefore exactly $g_{uv}=0$. Integrating first in $u$ and then in $v$ on a coordinate rectangle giveswhere $A,B$ are arbitrary twice continuously differentiable functions. The Jacobian determinant of the transformation is $-2u/v$, so this calculation uses patches with $u,v\ne 0$. On a positive-quadrant patch we can take $u=\sqrt{xy}$ and $v=\sqrt{x/y}$, obtainingChanging branches merely redefines the arbitrary functions.

For completeness, the original equation also makes sense in quadrants with $xy<0$, where that particular real square-root transformation is unavailable. Multiplying the equation by $x$ givesOn any quadrant with $xy\ne 0$, put $a=\log |x|$, $b=\log |y|$; then set $\xi =a+b$, $\eta =a-b$. The equation becomes $4f_{\xi \eta }=0$. Thus an equivalent general local answer on every such quadrant isHere $\Phi ,\Psi$ are arbitrary $C^2$ functions on the relevant intervals. If a solution is required across $y=0$, its quadrant expressions must extend with matching derivatives; on that line the equation itself imposes $x{f}_{xx}(x,0)+f_x(x,0)=0$, so its trace is $a_0\log |x|+b_0$ on each interval with $x\ne 0$. The displayed coordinate transformations do not assert invertibility at $y=0$.