Math 6410 • ODE

Consider the first-order equation
\[
  u=x_1u_{x_1}+x_2u_{x_2}+\frac12\left(u_{x_1}^2+u_{x_2}^2\right),
\]
for the function \(u(x_1,x_2)\).
\begin{enumerate}
\item[(a)] Write down the characteristic equations
\[
\begin{aligned}
  \dot{p} &= -D_xF-D_zF\,p,\\
  \dot{z} &= D_pF\cdot p,\\
  \dot{x} &= D_pF,
\end{aligned}
\]
associated with this problem.
\item[(b)] Verify that the Cauchy data
\(u(x_1,0)=\frac12(1-x_1^2)\), \(x_1\in\R\), is noncharacteristic, and find an
\textit{admissible} set of initial conditions.
\item[(c)] Find a solution \(u(x_1,x_2)\) to the original equation which satisfies the
given Cauchy data.
\end{enumerate}