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.
Is there a name for $x(x-1)y''+[(1+c_1+c_2)x-c_3]y'+c_1c_2y=0$?
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ordinary-differential-equations
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0I'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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0See 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.