dsolve fails to integrate simple ODE
Oscar Benjamin
I'm not sure if I'm missing a trick but I've been trying to use dsolve and it seems it doesn't work in many simple cases. I've put some examples below, tested with sympy 1.2 installed using pip. I don't know if any of the below is me using dsolve incorrectly or should be considered a bug or is just a reflection of it being a work in progress.
I want to solve the heat/diffusion equation for heat in a cylinder with an exponential heat term:
$ isympy
In [1]: eqn = Derivative(Derivative(f(x),x)*x,x)/x - exp(x)
In [2]: eqn
Out[2]:
d ⎛ d ⎞
──⎜x⋅──(f(x))⎟
x dx⎝ dx ⎠
- ℯ + ──────────────
x
In [3]: dsolve(eqn, f(x))
---------------------------------------------------------------------------
NotImplementedError
In [16]: res = expand((simplify(eqn * x).integrate(x) + C1)/x).integrate(x) + C2
...:
In [17]: res
Out[17]:
x
C₁⋅log(x) + C₂ + f(x) - ℯ + Ei(x)
In [18]: solve(res, f(x))
Out[18]:
⎡ x ⎤
⎣-C₁⋅log(x) - C₂ + ℯ - Ei(x)⎦
In [23]: eqn = log(log(f(x))*x)/x - exp(x)
In [24]: eqn
Out[24]:
x log(x⋅log(f(x)))
- ℯ + ────────────────
x
In [25]: solve(eqn, f(x))
Out[25]:
⎡ x⎤
⎢ x⋅ℯ ⎥
⎢ ℯ ⎥
⎢ ─────⎥
⎢ x ⎥
⎣ℯ ⎦
In [26]: eqn = ((f(x)).diff(x)*x).diff(x)/x - exp(x)
In [27]: eqn
Out[27]:
2
d d
x⋅───(f(x)) + ──(f(x))
2 dx
x dx
- ℯ + ──────────────────────
x
In [28]: dsolve(eqn)
---------------------------------------------------------------------------
NotImplementedError
In [32]: eqn = (g(x)*x).diff(x)/x - exp(x)
In [33]: eqn
Out[33]:
d
x⋅──(g(x)) + g(x)
x dx
- ℯ + ─────────────────
x
In [34]: dsolve(eqn, g(x))
Out[34]:
x
C₁ x ℯ
g(x) = ── + ℯ - ──
x x
In [41]: dsolve(eqn, g(x)).integrate(x)
Out[41]:
⌠
⎮ ⎛ x⎞
⌠ ⎮ ⎜C₁ x ℯ ⎟
⎮ g(x) dx = ⎮ ⎜── + ℯ - ──⎟ dx
⌡ ⎮ ⎝x x ⎠
⌡
In [40]: dsolve(eqn, g(x)).rhs.integrate(x)
Out[40]:
x
C₁⋅log(x) + ℯ - Ei(x)
Aaron Meurer
equation can be solved by integrating as you mentioned. There is an
issue to implement this algorithm, but it hasn't been done yet
https://github.com/sympy/sympy/issues/6259.
In expanded form it is a Euler equation, but the algorithm isn't smart
enough to recognize it (you need to multiply through by x**2). If you
do that you get an answer:
>>> dsolve((eqn.doit()*x**2).expand())
Eq(f(x), C1 + C2*log(x) + (x - 1)*exp(x)*log(x) - Integral(x*exp(x)*log(x), x))
I opened https://github.com/sympy/sympy/issues/15217 for this.
I guess based on your other calculation the answer should be
expressible via Ei, which SymPy doesn't know how to do for that
integral.
Regarding the final issue, I'm not sure why integrate() on an Eq
doesn't evaluate the integral, but if you call doit() it computes it.
>>> dsolve(eqn, g(x)).integrate(x).doit()
Eq(Integral(g(x), x), C1*log(x) + exp(x) - Ei(x))
I opened https://github.com/sympy/sympy/issues/15218 for this.
Aaron Meurer
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Oscar Benjamin
In [43]: eqn = exp(f(x).diff(x)-f(x))
In [44]: eqn
Out[44]:
d
-f(x) + ──(f(x))
dx
ℯ
In [45]: dsolve(eqn, f(x))
---------------------------------------------------------------------------
IndexError Traceback (most recent call last)
<ipython-input-45-b99728060ab1> in <module>()
----> 1 dsolve(eqn, f(x))
~/current/sympy/venv/lib/python3.6/site-packages/sympy/solvers/ode.py in dsolve(eq, func, hint, simplify, ics, xi, eta, x0, n, **kwargs)
662 # The key 'hint' stores the hint needed to be solved for.
663 hint = hints['hint']
--> 664 return _helper_simplify(eq, hint, hints, simplify, ics=ics)
665
666 def _helper_simplify(eq, hint, match, simplify=True, ics=None, **kwargs):
~/current/sympy/venv/lib/python3.6/site-packages/sympy/solvers/ode.py in _helper_simplify(eq, hint, match, simplify, ics, **kwargs)
687 # attempt to solve for func, and apply any other hint specific
688 # simplifications
--> 689 sols = solvefunc(eq, func, order, match)
690 if isinstance(sols, Expr):
691 rv = odesimp(sols, func, order, cons(sols), hint)
~/current/sympy/venv/lib/python3.6/site-packages/sympy/solvers/ode.py in ode_lie_group(eq, func, order, match)
5460 else:
5461 y = Dummy("y")
-> 5462 h = sol[0].subs(func, y)
5463
5464 if xis is not None and etas is not None:
IndexError: list index out of range
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Aaron Meurer
Aaron Meurer
On Tue, Sep 11, 2018 at 5:28 PM, Oscar Benjamin
Oscar Benjamin
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