1. Aug 27, 2011

### Lawrencel2

Show that under the transformation

u=x, v=$\alpha$x+$\beta$t​
Ayxx+Byxt+Cytt=0 ; B^2-4AC>0
becomes

Ayuu+(2A$\alpha$+B$\beta$)yuv+(A$\alpha$2+B$\alpha$$\beta$+C$\beta$2)yvv=0

(A,B,C are constants)
I have no idea where to start. and i have to present this problem to the front of my class on monday. Can anyone give me a big head start or anything?

2. Aug 27, 2011

### jackmell

gotta' start taking partials. That's where to start.

Is it like a bunch of people in there?

Ok, just playing.

So if y=f(x,t) and x=u and v=ax+bt, then:

$$y_x=y_u u_x+y_v v_x$$
$$y_x=y_u+a y_v$$

but the second one is a litle tricky since you taking the partial of partials so:

$$y_{xx}=\frac{\partial}{\partial x} \left(y_u+a y_v\right)=y_{uu} u_x+y_{vu} v_x+a\left(y_{uv}u_x+y_{vv} v_x\right)$$

Ok then, keep doing that for each partial in the first expression, make all those substitutions back into the first expression which wil turn it into an expression of y in terms of u and v, then simplify, cancel, whatever, and it should look like the second expression.