 Solution

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.