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

1. Jan 12, 2009

### jianxu

1. The problem statement, all variables and given/known data

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$$\delta$$u/$$\delta$$t + 3$$\delta$$u/$$\delta$$x = 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?

3. The attempt at a solution

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

Thank you

2. Jan 12, 2009

### Dick

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

3. Jan 12, 2009

### jianxu

I understand what that means but the book seems to want a different way to approach this. In their example, they used $$\delta$$u/$$\delta$$t + $$\delta$$u/$$\delta$$x = 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:

$$\delta$$u/$$\delta$$x=$$\delta$$u/$$\delta$$$$\alpha$$*$$\delta$$$$\alpha$$/$$\delta$$x + $$\delta$$u/$$\delta$$$$\beta$$*$$\delta$$$$\beta$$/$$\delta$$t

and then they did the same for $$\delta$$u/$$\delta$$t

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

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