= Solution
The <characteristic equation of a partial differential equation> for $u_{xx}+u_{xy}=0$ is $(dy)^2-dx\,dy=0$. Thus the two <characteristic curve> families are $\boxed{y=\text{constant},\quad y-x=\text{constant}}$. 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 to
$$
\boxed{U_{\xi\eta}=0.}
$$
Integrating 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 $-G'(y)=e^y$. Hence $G(y)=-e^y+C$ and $F(y)=e^y-C$, so
$$
\boxed{u(x,y)=e^y-e^{y-x}.}
$$
Direct 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.
Back to article page
