Toronto Math Forum

Based on Tristan's work,
\begin{eqnarray}
u(x,y)&=&\phi(y)x+\psi(y)\\
u_{yy}&=&0
\end{eqnarray}
Consider derive $u(x,y)$ with respect to y:
\begin{eqnarray}
u_y&=&\phi'(y)x+\psi'(y)\\
u_{yy}&=&\phi''(y)x+\psi''(y)
\end{eqnarray}
Then plug (5) into (7):
\begin{eqnarray}
\phi''(y)x=-\psi''(y)
\end{eqnarray}
Integrating both sides with respect to y:
\begin{eqnarray}
x(\phi'(y)+g(x))=-\psi'(y)+f(x)
\end{eqnarray}
Let me rearrange it into:
\begin{eqnarray}
x\phi'(y)+\psi'(y)=f(x)-xg(x)
\end{eqnarray}
Then plug (10) into (6):
\begin{eqnarray}
u_y=f(x)-xg(x)
\end{eqnarray}
Integrate it with respect to y, we get:
\begin{eqnarray}
u(x,y)=-xyg(x)+yf(x)+p(x)
\end{eqnarray}