def match_riccati(eq, f, x):
"""
A function that matches and returns the coefficients
if an equation is a Riccati ODE
Parameters
==========
eq: Equation to be matched
f: Dependent variable
x: Independent variable
Returns
=======
match: True if equation is a Riccati ODE, False otherwise
funcs: [b0, b1, b2] if match is True, [] otherwise. Here,
b0, b1 and b2 are rational functions which match the equation.
"""
# Group terms based on f(x)
if isinstance(eq, Eq):
eq = eq.lhs - eq.rhs
eq = eq.expand().collect(f(x))
cf = eq.coeff(f(x).diff(x))
# There must be an f(x).diff(x) term.
# eq must be an Add object since we are using the expanded
# equation and it must have atleast 2 terms (b2 != 0)
if cf != 0 and isinstance(eq, Add):
# Divide all coefficients by the coefficient of f(x).diff(x)
# and add the terms again to get the same equation
eq = Add(*((x / cf).cancel() for x in eq.args)).collect(f(x))
# Match the equation with the pattern
b1 = -eq.coeff(f(x))
b2 = -eq.coeff(f(x)**2)
b0 = (f(x).diff(x) - b1 * f(x) - b2 * f(x)**2 - eq).expand()
funcs = [b0, b1, b2]
# Check if coefficients are not symbols and floats
if any(len(x.atoms(Symbol)) > 1 or len(x.atoms(Float)) for x in funcs):
return False, []
# If b_0(x) contains f(x), it is not a Riccati ODE
if len(b0.atoms(f)) or not all(
(b2 != 0, b0.is_rational_function(x), b1.is_rational_function(x),
b2.is_rational_function(x))):
return False, []
return True, funcs
return False, []
