Modify an expression involving derivatives of functions by changing variables - sympy

Question

I am attempting to make a YouTube video in which I demonstrate how SymPy can be used to verify the steps in the derivation of a solution to Schrodinger's equation. There is an expression involving derivatives"
u=symbols('u')(r)
diff(u,r,2)
and a change of variables (a is constant):
r=a*x
or
u.subs(r,a*x)
and I want to get back something like:
diff(u,x,2)/a**2
I understand that I may need something like
u=Function('u')(x)
or
u=Symbol('u', class=Function)
But I have already defined u:
u=Function('u')(r)
I have been unable to get anything to work. If I differentiate u with respect to x, I get zero because (apparently) SymPy sees u as depending on r but not x. I do not understand the Sympy Function class.
This works for built-in functions like cos:
diff(cos(a*x),x)
returns
-a*sin(a*x)
I just want the same behavior for my abstract function u.

Answer 1

Substituting for the independent variable in a differential equation is something that is difficult or not yet properly supported in sympy. You can do this:
In [9]: u = Function('u')
In [10]: r = Symbol('r')
In [11]: u(r)
Out[11]: u(r)
In [12]: u(r).diff(r)
Out[12]:
d
──(u(r))
dr
In [13]: a, x = symbols('a, x')
In [15]: u(r).diff(r).subs(r, a*x)
Out[15]:
⎛d ⎞│
⎜──(u(r))⎟│
⎝dr ⎠│r=a⋅x
Part of the problem is that sympy doesn't have a way to represent u' as the derivative of the single argument function u without differentiating with respect to a symbol.
What is needed is something like dchange from Maple:
https://github.com/sympy/sympy/issues/17590
There are complicated ways to make this work in sympy if needed but those are probably not suitable for a youtube video. I suggest to skip over this section in the video :)
In [1]: u = Function('u')
In [2]: a, r, x = symbols('a, r, x')
In [3]: expr = u(r).diff(r, 2)
In [4]: expr
Out[4]:
2
d
───(u(r))
2
dr
In [5]: expr.subs(u(r).diff(r, 2), 1/a**2 * u(x).diff(x, 2)) # cheating
Out[5]:
2
d
───(u(x))
2
dx
─────────
2
a