#!/usr/bin/env python
# coding: utf-8

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from sympy import *


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init_session()


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r, theta, phi, kappa, alpha = symbols('r theta phi kappa alpha')
R, Theta = symbols('R Theta', cls=Function)
u = f(r, theta)


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eq = 1/r**2*diff(r**2*u.diff(r), r) + 1/(r**2*sin(theta))*diff(sin(theta)*u.diff(theta), theta) + kappa**2*u


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aux = expand(r**2*eq.subs(f(r, theta), R(r)*Theta(theta)).doit()/(R(r)*Theta(theta)))


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aux


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eq1 = Theta(theta).diff(theta, 2) + cos(theta)/sin(theta)*Theta(theta).diff(theta) - alpha**2*Theta(theta)
eq1


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eq2 = expand(aux - eq1/Theta(theta))
eq2 = expand(eq2*R(r))
eq2


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print(eq2)


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