Sympy simple second order ode

Question

I want to perform ode with Sympy.

If I start with a simple one, such as, f''(x) = f(x), dsolve works fine

import sympy as sym

z = sym.symbols('z', real=True) 
Phi = sym.Function('Phi')(z)
Phi_ = sym.Derivative(Phi,z)
Phi__ = sym.Derivative(Phi_,z)

Eqn1 = sym.Eq(Phi__, Phi) # f'' = f

sol1 = sym.dsolve(Eqn1) # Find solution

However, I try this ode f''(x) = exp(f(x)) (The solution can be check with Wolfram Alpha.)

Eqn2 = sym.Eq(Phi__, sym.exp(Phi))  # f'' = exp (f)
sol2 = sym.dsolve(Eqn2) # ERROR

I have an error:

NotImplementedError: solve: Cannot solve -exp(Phi(z)) + Derivative(Phi(z), (z, 2))

Is this a limitation of Sympy? Should I use another function?

Answer 1

I don't know what version of SymPy you are using but I'm using 1.9 and I just checked all versions back to 1.2 and this ODE is always solved:

In [1]: import sympy as sym
   ...: 
   ...: z = sym.symbols('z', real=True)
   ...: Phi = sym.Function('Phi')(z)
   ...: Phi_ = sym.Derivative(Phi,z)
   ...: Phi__ = sym.Derivative(Phi_,z)
   ...: 
   ...: Eqn1 = sym.Eq(Phi__, Phi) # f'' = f
   ...: 
   ...: sol1 = sym.dsolve(Eqn1) # Find solution

In [2]: sol1
Out[2]: 
           -z       z
Φ(z) = C₁⋅ℯ   + C₂⋅ℯ