Math 6410 • ODE

Consider the equation
\[
  a(x,y)u_x+b(x,y)u_y=c(x,y,u)
\]
where \(a\), \(b\) and \(c\) are suitably smooth.
\begin{enumerate}
\item Define what is meant by an integral surface for the above equation.
\item Write down the characteristic equations and show how they can be used to construct
an integral surface that contains the curve
\(x=\phi(s)\), \(y=\psi(s)\), \(u=\eta(s)\), where \(\phi\), \(\psi\) and \(\eta\) are \(C^1\).
\item What additional conditions, if any, must be placed on the data to guarantee a
locally unique solution? Explain carefully.
\item Demonstrate the above method by solving
\[
\begin{aligned}
  u_x+2u_y&=-u,\\
  u(0,y)&=f(y)
\end{aligned}
\]
where \(f\) is of class \(C^1\).
\end{enumerate}