Math 6410 • ODE

Suppose that the initial value problem for the wave equation
\[
  \begin{aligned}
    \frac{\partial^2u}{\partial t^2}-c^2\nabla^2u&=0,
      &&x\in\R^n,\quad t>0,\\
    u(0,x)&=g(x),\\
    \frac{\partial u}{\partial t}(0,x)&=0,
      &&x\in\R^n
  \end{aligned}
\]
has a solution \(u_g\). Find (and verify) the solution \(u_h\) of the same equation
subject to the initial conditions
\[
  \begin{aligned}
    u(0,x)&=g(x),\\
    \frac{\partial u}{\partial t}(0,x)&=h(x),
      &&x\in\R^n
  \end{aligned}
\]
in terms of \(u_g\), \(g\) and \(h\).