Math 6410 • ODE

The quasilinear Cauchy problem
\[
xu_x+yuu_y+xy=0,\qquad u=5\quad\text{when }xy=1,
\]
has the solution
\[
u(x,y)=-1+\sqrt{38-2xy}.
\]

\begin{enumerate}
\item[(a)] Derive this solution by the method of characteristics. (Hint: Along a
characteristic \(x(s)\), \(y(s)\), \(u(s)\) find \(d(xy)/ds\) and then \(du/dz\) where
\(z=xy\).)

\item[(b)] Verify that \(u(x,y)\) is a solution in the plane domain defined by
\(xy<19\).
\end{enumerate}