Reduction of PDE to an ODE by means of linear change of variables

[jianxu]

Homework Statement

So it's been a really long time since I've done any ode/linear algebra and would like some help with this problem.

Derive the general solution of the given equation by using an appropriate change of variables

2\deltau/\deltat + 3\deltau/\deltax = 0

The thing that I'm really confused about is how do we decide what an appropriate change of variable is? Is there a general rule that I should go by?

The Attempt at a Solution

none yet because I'm not sure how to find the appropriate change of variable

Thank you

 

[Dick]

(2*d/dt+3*d/dx)u=0. Suppose u is a function only of 3t-2x?

 

[jianxu]

(2*d/dt+3*d/dx)u=0. Suppose u is a function only of 3t-2x?

I understand what that means but the book seems to want a different way to approach this. In their example, they used \deltau/\deltat + \deltau/\deltax = 0.

Next, they made two linear change of variables equations i think...

\alpha = ax+bt and \beta = cx + dt
where a,b,c,d will be chosen appropriately. Then they used chain rule in 2 dimension giving:

\deltau/\deltax=\deltau/\delta\alpha*\delta\alpha/\deltax + \deltau/\delta\beta*\delta\beta/\deltat

and then they did the same for \deltau/\deltat

which gives them (a+b)\deltau/\delta\alpha+(c+d)\deltau/\delta\beta = 0

Then they assumed a =1, b=0, c=1, d=-1 which gives them:
\deltau/\delta\alpha=0

and they were able to find the general solution from there.

So I guess what I meant is how were they able to determine what \alpha and \beta are?