= Solution
The first-order <chain rule>, with $x,y$ regarded as functions of $u,v$, is
$$
g_u=x_uf_x+y_uf_y,\qquad g_v=x_vf_x+y_vf_y.
$$
Apply 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 gives
$$
g_{uv}=x_ux_vf_{xx}+(x_uy_v+x_vy_u)f_{xy}
+y_uy_vf_{yy}+x_{uv}f_x+y_{uv}f_y.
$$
\b[Thus $\boxed{H=x_{uv},\ K=y_{uv}}$.] These terms record the second derivatives of the coordinate transformation, not of $f$.
For the given transformation,
$$
x_u=v,\quad x_v=u,\quad y_u=\frac1v,\quad y_v=-\frac u{v^2},
\quad x_{uv}=1,\quad y_{uv}=-\frac1{v^2}.
$$
The mixed-derivative coefficient is zero, while $x_ux_v=x$, $y_uy_v=-y^2/x$, 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 gives
$$
\boxed{g(u,v)=A(u)+B(v),}
$$
where $A,B$ are arbitrary twice continuously differentiable functions. The <Jacobian determinant> of the transformation is $-2u/v$, so this calculation uses patches with $u,v\ne0$. On a positive-quadrant patch we can take $u=\sqrt{xy}$ and $v=\sqrt{x/y}$, obtaining
$$
\boxed{f(x,y)=A(\sqrt{xy})+B(\sqrt{x/y}).}
$$
Changing 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 $x$ gives
$$
(x\partial_x)^2f-(y\partial_y)^2f=0.
$$
On any quadrant with $xy\ne0$, 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 is
$$
\boxed{f(x,y)=\Phi(\log|xy|)+\Psi(\log|x/y|).}
$$
Here $\Phi,\Psi$ 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 $xf_{xx}(x,0)+f_x(x,0)=0$, so its trace is $a_0\log|x|+b_0$ on each interval with $x\ne0$. The displayed coordinate transformations do not assert invertibility at $y=0$.
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