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Is there a name for this differential equation? $x(x-1)y''+[(1+c_1+c_2)x-c_3]y'+c_1c_2y=0$ Thanks.

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    What about it: http://www.wolframalpha.com/input/?i=x%28x-1%29y%27%27%2B%28%281%2Bc1%2Bc2%29x%2Bc3%29y%27+%2Bc1+c2+y+%3D02011-09-25
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    @Gortaur: Thanks, but Hmm... Is there a name there? I didn't mean name as in "2nd order ODE", but rather something like "Bessel's equation".2011-09-25
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    I understand that it wasn't the answer you expected, but at least we know that WA doesn't know it in its current formulation (I didn't know if you tried WA before asking here).2011-09-25
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    I don't know if it yields something useful, but have you tried putting it in Sturm-Liouville form? (Maybe WA or Mathematica can do this, saving you a lengthy calculation.)2011-09-25
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    @Gerben: Thanks, how can I get it in SL form using WA?2011-09-25
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    @Johnny: I can't find a quick fix, but [this site](http://www.math.osu.edu/~gerlach/math/BVtypset/node63.html) explains how to do it by hand. You can use WA to do the integral $\exp[\int^x Q(t)/P(t) \mathrm{d}t].$2011-09-25
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    @Gerben: Thanks. I shall try that out. BTW, your suggestion seems to imply that the SL form is more commonly used. I know that the site teaches how to convert a 2nd ODE into that form, but why is this form preferred?2011-09-25
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    I'm not an ODE expert, but SL problems are a class of well-studied and understood differential equations, so if you manage to put an ODE in S-L form, you get a lot of properties about its spectrum and eigenfunctions for free.2011-09-26

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Maple classifies this as a Jacobi differential equation. The general solution is expressed in terms of hypergeometric functions: $y = a_{{1}}\ {{}_2F_1([c_{{1}},c_{{2}}],[c_{{3}}],\,x)}+a_{{2}}{x}^{1-c_{{3}}} \ {{}_2F_1([c_{{1}}+1-c_{{3}},c_{{2}}+1-c_{{3}}],[\,-c_{{3}}+2],\,x)}$ where $a_1$ and $a_2$ are arbitrary constants.

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    Or, since it's a Jacobi DE after all, the solution is expressible in terms of Jacobi polynomials and Jacobi functions of the second kind.2011-09-26
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    Jacobi polynomials if $c_1$ is a nonpositive integer, I think. Otherwise not polynomials.2011-09-26
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    I suppose in that case, we use "Jacobi functions of the first kind" (by analogy with the Legendre case).2011-09-26
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    How does you feel that this is a Jacobi differential equation? According to http://mathworld.wolfram.com/JacobiDifferentialEquation.html, note that the $y''$ term of a Jacobi differential equation is $1-x^2$ while the $y''$ term of a Gaussian hypergeometric equation is $x(x-1)$ .2012-10-15
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    See http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor%2FJacobi An affine transformation of the independent variable takes you from $1-x^2$ to $x(1-x)$.2012-10-15
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This is exactly a Gaussian hypergeometric equation.

http://eqworld.ipmnet.ru/en/solutions/ode/ode0222.pdf