Hello!

I am working on a question involving a P.D.E. in a adv. calc. course.

Consider the general homogenous second-order partial differential equation

$\displaystyle a\frac{\delta^2u}{\delta x^2} + 2b\frac{\delta^2 u}{\delta x\delta y} + c\frac{\delta^2 u}{\delta y^2} = 0 $

with constant coefficients (a,b,c).

If $\displaystyle ac - b^2 = 0$, show that the substitution $\displaystyle s = bx - ay, \ t = y$ reduces the above equation to $\displaystyle \frac{\delta^2 u}{\delta t^2} = 0$

So we must consider the change of variable from $\displaystyle u(x,y)$ to $\displaystyle u(s,t)$, which is $\displaystyle u(bx-ay, y)$ and take partial derivatives? I can't seem to derive the desired result, however, and I have double-checked my messy algebra...

Any help appreciated. Thanks!