Differential equation change of variables with sympy

I have an ordinary differential equation like this:

DiffEq = Eq(-ℏ*ℏ*diff(Ψ,x,2)/(2*m) + m*w*w*(x*x)*Ψ/2 - E*Ψ   ,  0)

I want to perform a variable change :

sp.Eq(u , x*sqrt(m*w/ℏ))
sp.Eq(Ψ, H*exp(-u*u/2))

How can I do this with sympy?

Answer

Use the following function:

def variable_change(ODE,dependent_var, 
                    independent_var,
                    new_dependent_var = None, 
                    new_independent_var= None, 


                    dependent_var_relation = None,
                    independent_var_relation = None,
                    order = 2):





    if new_dependent_var == None:
        new_dependent_var = dependent_var
    if new_independent_var == None:
        new_independent_var = independent_var




    # dependent variable change

    if new_independent_var != independent_var:

        for i in range(order, -1, -1):

            # remplace derivate
            a = D(dependent_var , independent_var, i )
            ξ = Function("ξ")(independent_var)

            b = D( dependent_var.subs(independent_var, ξ),  independent_var  ,i)

            rel = solve(independent_var_relation, new_independent_var)[0]


            for j in range(order, 0, -1):
                b = b.subs( D(ξ,independent_var,j), D(rel,independent_var,j))

            b = b.subs(ξ, new_independent_var)

            rel = solve(independent_var_relation, independent_var)[0]
            b = b.subs(independent_var, rel)


            ODE =   ODE.subs(a,b)

        ODE = ODE.subs(independent_var, rel)


    # change of variables of indpendent variable


    if new_dependent_var != dependent_var:

        ODE = (ODE.subs(dependent_var.subs(independent_var,new_independent_var) , (solve(dependent_var_relation, dependent_var)[0])))
        ODE = ODE.doit().expand()

    return ODE.simplify()

For the example posted:

from sympy import *
from sympy import diff as D

E, ℏ ,w,m,x,u = symbols("E, ℏ , w,m,x,u")
Ψ ,H = map(Function, ["Ψ ","H"])
Ψ ,H = Ψ(x), H(u)



DiffEq = Eq(-ℏ*ℏ*D(Ψ,x,2)/(2*m) + m*w*w*(x*x)*Ψ/2 - E*Ψ,0)
display(DiffEq)



display(Eq(u , x*sqrt(m*w/ℏ)))
display(Eq(Ψ, H*exp(-u*u/2)))


newODE = variable_change(ODE = DiffEq,


                independent_var = x, 
                new_independent_var= u,
                independent_var_relation = Eq(u , x*sqrt(m*w/ℏ)),
                dependent_var = Ψ,  


                new_dependent_var = H,   
                dependent_var_relation = Eq(Ψ, H*exp(-u*u/2)),

                order = 2)







display(newODE)

Under this substitution the differential equation outputted is then:

Eq((-E*H + u*w*ℏ*D(H, u) + w*ℏ*H/2 - w*ℏ*D(H, (u, 2))/2)*exp(-u**2/2), 0)