CODEWITHSUNDEEP
pythonsympyode
ty_743
ty_743

Reputation: 1

How to solve separable differential equation using Sympy?

I cannot figure out how to solve this separable differential equation using sympy. Help would be greatly appreciated.

y′=(y−4)(y−2),y(0)=5

Here was my attempt, thanks in advance!!!

import sympy as sp


x,y,t = sp.symbols('x,y,t')

y_ = sp.Function('y_')(x)

diff_eq = sp.Eq(sp.Derivative(y_,x), (y-4)*(y-2))

ics = {y_.subs(x,0):5}

sp.dsolve(diff_eq, y_, ics = ics)

the output is y(x) = xy^2 -6xy +8x + 5

Upvotes: 0

Views: 88

Answers (1)

Lutz Lehmann
Lutz Lehmann

Reputation: 25992

The primary error is the introduction of y_. This makes the variable y a constant parameter of the ODE and you get the wrong solution.

If you correct this you get an error of "too many solutions for the integration constant". This is a bug caused by not simplifying the integration constant after it first occurs. So multiplication and addition of constants should just be absorbed, an additive constant in an exponent should become a multiplicative factor for the exponential. As it is, exp(2*C_1)==3 has two solutions if C_1 is considered as an angle (it's a bit of tortured logic from computing roots in the complex plane).

The newer versions can actually solve this fully if you give the third hint in the classification list 'separable', '1st_exact', '1st_rational_riccati', ... that does something different than partial fraction decomposition of the first two

from sympy import *
x = Symbol('x')
y = Function('y')(x)

dsolve(Eq(y.diff(x), (y-2)*(y-4)),y,
       ics={y.subs(x,0):5}, 
       hint='1st_rational_riccati')

returning

\displaystyle y{\left(x \right)} = \frac{2 \cdot \left(6 - e^{2 x}\right)}{3 - e^{2 x}}

solution formula

Upvotes: 1

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