Sympy: solve a differential equation

Question

I want to find an elegant way of solving the following differential equation:

from sympy import *
init_printing()

M, phi, t, r = symbols('M phi t r')

eq = Eq(-M * phi(t).diff(t), Rational(3, 2) * m * r**2 * phi(t).diff(t) * phi(t).diff(t,t))

I assume that phi(t).diff(t) is not zero. Hence the left and right side are shortened.

This is how I get to the solution:

# I assume d/dt(phi(t)) != 0

theta = symbols('theta')
eq = eq.subs({phi(t).diff(t, 2): theta})  # remove the second derivative
eq = eq.subs({phi(t).diff(t): 1})  # the first derivative is shortened
eq = eq.subs({theta: phi(t).diff(t, 2)})  # get the second derivative back

dsolve(eq, phi(t))

How do I solve this more elegantly?

Answer 1

Ideally dsolve() would be able to solve the equation directly, but it doesn't know how (it needs to learn that it can factor an equation and solve the factors independently). I opened an issue for it.

My only other suggestion is to divide phi' out directly:

eq = Eq(eq.lhs/phi(t).diff(t), eq.rhs/phi(t).diff(t))

You can also use

eq.xreplace({phi(t).diff(t): 1})

to replace the first derivative with 1 without modifying the second derivative (unlike subs , xreplace has no mathematical knowledge of what it is replacing; it just replaces expressions exactly).

And don't forget that phi(t) = C1 is also a solution (for when phi' does equal 0).