Using sympy for ODEs

We can use the capacity in sympy to differentiate symbolic expressions for simple verification of solutions of an ODE or PDE. For ODEs, there is an extensive set of documentation that deals with finding sets of solutions:

Often, however, we are verifying a solution we know or suspect to have a certain form and we simply need sympy to make sure there are no mistakes.

Let’s begin with a simple harmonic oscillator:

\[ \frac{d^2 \theta}{d t^2} = -k \theta \]

which we expect to have solutions like

\[ \theta = A \cos(\omega t + \phi) \]

Can we verify these solutions using symbolic manipulation ?

import sympy
import math
import numpy as np

Symbolic approach

from sympy.core.symbol import Symbol

t = Symbol('t')
A = Symbol('A')

omega = Symbol('omega', positive=True)
phi = Symbol('phi')
theta = A * sympy.cos(omega * t + phi)

Let’s now check to see whether this form of theta is an eigenfunction of the ODE

theta.diff(t,2) / theta

So, yes, this satisfies the equation subject to additional information needed to determine \(\phi\). The value of \(\omega\) is \(\sqrt{k}\).

Use of the sympy equation module (Eq)

If we tell sympy that we have an equation, there are tools we can use to solve it

Theta = sympy.Function("Theta")

eq=sympy.Eq(Theta(t).diff(t,2) + omega**2 * Theta(t), 0)
c1,c2 = list(sol.free_symbols)[0], list(sol.free_symbols)[1]
display(c1, c2)
myform = A * sympy.cos(omega * t + phi)
their_form = sol.subs([(c1, -A * sympy.sin(phi)), (c2, A * sympy.cos(phi))])
(myform - their_form).simplify()
# Check to see if they are (exactly) equivalent