6 7.36, §6 Alternate

An alternate solution procedure is to define a new unknown:

$\begin{displaymath} v \equiv u_y - p u \end{displaymath}$

You must derive the problem for v:

The boundary condition is simply:

$\begin{displaymath} v(x,0) = f(x) \end{displaymath}$

To get the PDE for , use

$\begin{displaymath} \frac{\partial [PDE]}{\partial y} - p [PDE] \quad\Longrightarrow\quad v_{tt} = a^2 v_{xx} \end{displaymath}$

Similarly, for the initial conditions:

$\begin{displaymath} \frac{\partial [ICs]}{\partial y} - p [ICs] \quad\Longrightarrow\quad v(0,y) = v_x(0,y) = 0 \end{displaymath}$

**Figure 6:** Problem for v.
$\begin{figure}\begin{center} \leavevmode \epsffile{figures/5.ps} \end{center}\end{figure}$

After finding , I still need to find from the definition of :

$\begin{displaymath} v \equiv u_y - p u \end{displaymath}$

Where do you get the integration constant??