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The characteristic equation of a partial differential equation for $u_{xx}+u_{xy}=0$ is $(dy{)}^2-dxdy=0$. Thus the two characteristic curve families are $\boxed{y=\text{constant},y-x=\text{constant}}$. Set $\xi =y$ and $\eta =y-x$. The chain rule gives ${\partial }_x=-{\partial }_{\eta }$ and ${\partial }_y={\partial }_{\xi }+{\partial }_{\eta }$, reducing the equation toIntegrating in the two independent characteristic coordinates gives $U=F(\xi )+G(\eta )$, or $u(x,y)=F(y)+G(y-x)$. At $x=0$, the value condition gives $F(y)+G(y)=0$, while the derivative condition gives $-G^{\prime }(y)=e^y$. Hence $G(y)=-e^y+C$ and $F(y)=e^y-C$, soDirect differentiation gives $u_x=e^{y-x}$, $u_{xx}=-e^{y-x}$ and $u_{xy}=e^{y-x}$, verifying both the equation and the two data. The additive constants cancel in $u$, so they do not create different solutions.

The principal coefficients are $A=-xy$, $B=0$, $C=1$, in the convention $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}$. Their discriminant is $B^2-AC=xy$. Thus it is a hyperbolic partial differential equation where $xy>0$ and an elliptic partial differential equation where $xy<0$. Its principal part has parabolic type where $xy=0$; however the full equation contains $u_y/y$ and is undefined on $y=0$. On its actual domain $y\ne 0$, the parabolic locus is $x=0$.

The characteristic equation of a partial differential equation isIn a hyperbolic region this gives two distinct real characteristic directions. The singular lower-order coefficient should not be mistaken for an ordinary parabolic point of the full equation on $y=0$.

In $x,y>0$, the characteristic equation of a partial differential equation gives $dx/\sqrt{x}=\pm \sqrt{y}dy$. Integration yields the characteristic coordinatesTheir Jacobian determinant is $-\sqrt{y}/(2\sqrt{x})\ne 0$, 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 givesSubstitution cancels the first-derivative terms, leaving $U_{ss}-U_{rr}=0$. With $\xi =r+s$, $\eta =r-s$ and $U=V(\xi ,\eta )$, the characteristic normal form is $V_{\xi \eta }=0$. Integrating once in each variable giveswhere $g,h$ are arbitrary twice continuously differentiable functions on the relevant coordinate intervals. This is a curvilinear reduction to the one-dimensional wave equation.