How to make Mathematica use the chain rule?

Question

Lets say I have the following PDE:

$$x^2 u_{xx} - u_{yy} + u_y = 0$$

And I have the following change of variables:

$$ s(x,y) = x e^y \, \, \, , \, \, t(x,y) = x e^{-y}$$

How can I use Mathematica to compute $u_{xx}, u_{yy},u_y$ on terms of $u_{ss},u_{tt}, u_{ts},u_t,u_s$ using the definitions and the chain rule?

Answer 1

expr = x^2 D[u[x, y], {x, 2}] - D[u[x, y], {y, 2}] + D[u[x, y], y]

$Assumptions = {s > 0, t > 0}
expr /. u -> (u[# Exp[#2], # Exp[-#2]] &) /. {x -> Sqrt[s t],
                                              y -> Log[Sqrt[s/t]]} // Simplify

---

Second set of replacement rules is from:

Eliminate[s == x Exp[y] && t == x Exp[-y], x]
Eliminate[s == x Exp[y] && t == x Exp[-y], y]

E^(2 y) t == s
s t == x^2 && (s t)/x == x

I've forced $Assumptions so things like Sqrt[s^2] are reduced to s but keep them in mind. Sometimes it may be important.

Answer 2

Just in case someone else need it (for example, quite possibly myself in the future), this is how I've done it

(Note: In the answer a different change of variables has been used)

s[x_, y_] := Log[x] + y
t[x_, y_] := Log[x] - y
Factor@D[u[s[x, y], t[x, y]], {x, 2}]
FullSimplify@D[u[s[x, y], t[x, y]], y]
Factor@D[u[s[x, y], t[x, y]], {y, 2}]
x^2 D[u[s[x, y], t[x, y]], {x, 2}] - 
  Factor@D[u[s[x, y], t[x, y]], {y, 2}] + 
  FullSimplify@D[u[s[x, y], t[x, y]], y] // FullSimplify

Which outputs the following:

Reading the output is quite anoying, but it was good enough for what I needed. (I'm sure it can be improved easily)