The algorithms to solve it aren't implemented. In factored form, the 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 On Tue, Sep 11, 2018 at 4:53 PM, Oscar Benjamin <[email protected]> wrote: > 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 > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/CAHVvXxROOENnFvfstx2ODdTC51JTYfaXgZMz%3DinO8HDKP2pSTA%40mail.gmail.com. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/CAKgW%3D6JpeWJJLZBPOgMnEFuRqsg3xc1-5gn6iiaDN9k07%2Bb4Dw%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout.
