MHB Need help with transforming one PDE to another

[Kane1]

For quick reference if you have the text, the question is from "Applied Partial Differential Equations" by J. David Logan. Section 1.9 #4
Show that the equation $$u_{tt} - c^2 u_{xx} + au_t + bu_x + du = f(x,t)$$
can be transformed into an equation of the form
$$w_{\xi\tau} + kw = g(\xi,\tau), w = w(\xi,\tau)$$
by first making the transformation $$\xi = x - ct, \tau = x + ct$$ and then letting $$u = w\exp{\alpha\xi + \beta\tau}$$ for some alpha and beta

I'm completely stumped at how to solve this problem, but I do not expect a solution, only a way to begin so I can get used to problems like these in general.

 

[Mathwebster]

Do you know how to transform first and second order derivatives under the change of variables

$\xi = x - ct$ and $\tau = x+ct$?

 

[Kane1]

I honestly have no idea. I know that you have to relate $u(x,t)$ with $w(\xi,\tau)$ and then use the chain rule. If you could help me with finding one derivative, like transforming the term $u_t$ I will try the others on my own. Thank you for your reply

 

[Mathwebster]

Here's the chain rule from Calc 3 (well, Calc 3 in US universities)

$u_t = u_{\tau} \tau_t + u_{\xi} \xi_t$

so if $\tau$ and $\xi$ are defined above, this becomes

$u_t = cu_{\tau} - c u_{\xi}$.

There are similar expressions for $u_x$, $u_{tt}$, $u_{tx}$ and $u_{xx}$.

 

[Kane1]

Here's the chain rule from Calc 3 (well, Calc 3 in US universities)

$u_t = u_{\tau} \tau_t + u_{\xi} \xi_t$

so if $\tau$ and $\xi$ are defined above, this becomes

$u_t = cu_{\tau} - c u_{\xi}$.

There are similar expressions for $u_x$, $u_{tt}$, $u_{tx}$ and $u_{xx}$.

Ah, I have seen that before in another class! I think I understand now. Thank you, thank you!