= Solution
In $x,y>0$, the <characteristic equation of a partial differential equation> gives $dx/\sqrt x=\pm\sqrt y\,dy$. Integration yields the <characteristic coordinates>
$$
\xi=\sqrt x+\frac{y^{3/2}}3,\qquad \eta=\sqrt x-\frac{y^{3/2}}3.
$$
Their <Jacobian determinant> is $-\sqrt y/(2\sqrt x)\ne0$, so they are valid independent coordinates. To see cancellation of the lower-order terms explicitly, first put $r=\sqrt x$, $s=y^{3/2}/3$ and $u(x,y)=U(r,s)$. The <chain rule> gives
$$
u_x=\frac{U_r}{2\sqrt x},\quad u_{xx}=\frac{U_{rr}}{4x}-\frac{U_r}{4x^{3/2}},\quad
u_y=\frac{\sqrt y}{2}U_s,\quad u_{yy}=\frac y4U_{ss}+\frac{U_s}{4\sqrt y}.
$$
Substitution cancels the first-derivative terms, leaving \b[$U_{ss}-U_{rr}=0$]. With $\xi=r+s$, $\eta=r-s$ and $U=V(\xi,\eta)$, the <characteristic normal form> is \b[$V_{\xi\eta}=0$]. Integrating once in each variable gives
$$
\boxed{u(x,y)=g\!\left(\sqrt x+\frac{y^{3/2}}3\right)+h\!\left(\sqrt x-\frac{y^{3/2}}3\right),}
$$
where $g,h$ are arbitrary twice continuously differentiable functions on the relevant coordinate intervals. This is a <curvilinear reduction to the one-dimensional wave equation>.
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