system of linear differential equations

[Michael Goerz]

I'm having a lot of problems trying to solve a system of linear differential

equations (which somehow looks like it should have an easy solution, but I might

be wrong):

    #!/usr/bin/env python

    import sympy

    α, β = sympy.symbols(r'\alpha, \beta', cls=sympy.Function)

    κ = sympy.symbols(r'\kappa', positive=True)

    t = sympy.symbols(r't', positive=True)

    c1 = κ / sympy.sqrt(2)

    c2 = κ * sympy.sqrt(2)

    DE1 = sympy.Eq(α(t).diff(t), c1 * β(t))

    DE2 = sympy.Eq(β(t).diff(t), -κ * β(t) - c2 * α(t))

    sympy.dsolve([DE1, DE2], [α(t), β(t)], hint='1st_linear')

    sympy.dsolve([DE1, DE2])

The first attempt gives me a backtrace because it fails to determine the order

for each function when the function names are given explicitly

    Traceback (most recent call last):

    File "./test_sympy.py", line 14, in <module>

        sympy.dsolve([DE1, DE2], [α(t), β(t)], hint='1st_linear')

    File "/Users/goerz/anaconda/envs/python34/lib/python3.4/site-packages/sympy/solvers/ode.py", line 579, in dsolve

        match = classify_sysode(eq, func)

    File "/Users/goerz/anaconda/envs/python34/lib/python3.4/site-packages/sympy/solvers/ode.py", line 1366, in classify_sysode

        if not order[func]:

    KeyError: \alpha(t)

This looks like a bug to me.

The second attempt gives me the following:

    Traceback (most recent call last):

    File "./test_sympy.py", line 13, in <module>

        sympy.dsolve([DE1, DE2])

    File "/Users/goerz/anaconda/envs/python34/lib/python3.4/site-packages/sympy/solvers/ode.py", line 612, in dsolve

        sols = solvefunc(match)

    File "/Users/goerz/anaconda/envs/python34/lib/python3.4/site-packages/sympy/solvers/ode.py", line 6394, in sysode_linear_2eq_order1

        sol = _linear_2eq_order1_type1(x, y, t, r)

    File "/Users/goerz/anaconda/envs/python34/lib/python3.4/site-packages/sympy/solvers/ode.py", line 6485, in _linear_2eq_order1_type1

        l1 = RootOf(l**2 - (r['a']+r['d'])*l + r['a']*r['d'] - r['b']*r['c'], 0)

    File "/Users/goerz/anaconda/envs/python34/lib/python3.4/site-packages/sympy/polys/rootoftools.py", line 70, in __new__

        raise PolynomialError("only univariate polynomials are allowed")

    sympy.polys.polyerrors.PolynomialError: only univariate polynomials are allowed

Not sure what to do with that.

I'm a little surprised this is so difficult. I was somehow assuming that this

would be a core feature (trying to replace Mathematica with SymPy).

The current documentation on systems of differential equations isn't overly

helpful :-/

Best,

Michael

[Aaron Meurer]

It looks like the bug has been fixed in SymPy master:

In [94]: print(sympy.dsolve([DE1, DE2], [α(t), β(t)]))
[Eq(\alpha(t), sqrt(2)*\kappa*(C1*sin(sqrt(3)*\kappa*t/2) +
C2*cos(sqrt(3)*\kappa*t/2))*exp(-\kappa*t/2)/2), Eq(\beta(t),
((-C1*\kappa/2 - sqrt(3)*C2*\kappa/2)*sin(sqrt(3)*\kappa*t/2) +
(sqrt(3)*C1*\kappa/2 -
C2*\kappa/2)*cos(sqrt(3)*\kappa*t/2))*exp(-\kappa*t/2))]

Aaron Meurer

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