Hi,
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
It strikes me as odd that sympy can't solve this since it's essentially an
algebraic rearrangement to get f here and all that is needed is to
understand that d/dx can be integrated. Sympy can do the integrals:
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)⎦
The algorithm for this is essentially the same as solving an algebraic
equation. If I replace the derivatives with logs then solve can do it:
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 ⎦
If I use f(x).diff(x) rather than Derivative then it looks more complicated
but should work:
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
The automatic expansion of the product rule here leaves us in a slightly
trickier position but again there is a general rule that sympy is
overlooking here: Use a substitution f' = g to bring it to 1st order:
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
Then strangely I get:
In [*41*]: dsolve(eqn, g(x)).integrate(x)
Out[*41*]:
⌠
⎮ ⎛ x⎞
⌠ ⎮ ⎜C₁ x ℯ ⎟
⎮ g(x) dx = ⎮ ⎜── + ℯ - ──⎟ dx
⌡ ⎮ ⎝x x ⎠
⌡
But if I just integrate the rhs it does the integral:
In [*40*]: dsolve(eqn, g(x)).rhs.integrate(x)
Out[*40*]:
x
C₁⋅log(x) + ℯ - Ei(x)
--
Oscar
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