pythonmatrixsympylinear-algebradifferential-equations

Solving a simple matrix differential equation

I want to use Sympy to solve the differential matrix equation: du/dt = [[0,1],[1,0]]u with initial value u(0) = [[4],[2]].

The answer is

enter image description here enter image description here

So the complete final answer is: 3e^t[[1],[1]] + e^{-t}[[1],[-1]]

How could I solve this with SymPy?

Solution

  • This can be achieved with dsolve_system:

    Code:

    from sympy import symbols, Eq, Function, init_printing
from sympy.solvers.ode.systems import dsolve_system

init_printing()

u1, u2 = symbols("u1 u2", cls=Function)
t = symbols("t")


eqs = [
    Eq(u1(t).diff(t), u2(t)),
    Eq(u2(t).diff(t), u1(t))
]

funcs = [u1(t), u2(t)]
solutions = dsolve_system(eqs, ics={u1(0): 4, u2(0): 2})
solutions[0]

    Output:

    enter image description here

    Output in latex:

    \left[ u_{1}{\left(t \right)} = 3 e^{t} + e^{- t}, \  u_{2}{\left(t \right)} = 3 e^{t} - e^{- t}\right]

    Edit For a general matrix, you can write this:

    import numpy as np

from sympy import symbols, Eq, Function, init_printing
from sympy.solvers.ode.systems import dsolve_system

init_printing()

# Replace with your matrix
A = np.array([
    [0, 1],
    [1, 0]
])
initial_condition = np.array([4, 2])

# Defines the symbols to be used
u = [Function(f'u{i}') for i in range(A.shape[0])]
t = symbols("t")

# Write the expression for the system of differential equations
eqs = []
for i, row in enumerate(A):
    rhs = 0
    for j, coeff in enumerate(row):
        rhs += coeff * u[j](t)

    eqs.append(Eq(u[i](t).diff(t), rhs))

# Solves the system of differential equations
ics = {u[i](0): val for i, val in enumerate(initial_condition)}
solutions = dsolve_system(eqs, ics=ics)

solutions[0]