Internal problem ID [10994]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 160.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\beta -\alpha -\left (\alpha +\beta +2\right ) x \right ) y^{\prime }+n \left (n +\alpha +\beta +1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 61
dsolve((1-x^2)*diff(y(x),x$2)+(beta-alpha-(alpha+beta+2)*x)*diff(y(x),x)+n*(n+alpha+beta+1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-n , n +\alpha +\beta +1\right ], \left [\beta +1\right ], \frac {x}{2}+\frac {1}{2}\right )+c_{2} \left (\frac {x}{2}+\frac {1}{2}\right )^{-\beta } \operatorname {hypergeom}\left (\left [-n -\beta , n +\alpha +1\right ], \left [1-\beta \right ], \frac {x}{2}+\frac {1}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.28 (sec). Leaf size: 69
DSolve[(1-x^2)*y''[x]+(\[Beta]-\[Alpha]-(\[Alpha]+\[Beta]+2)*x)*y'[x]+n*(n+\[Alpha]+\[Beta]+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\alpha } c_2 (x-1)^{-\alpha } \operatorname {Hypergeometric2F1}\left (-n-\alpha ,n+\beta +1,1-\alpha ,\frac {1-x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (-n,n+\alpha +\beta +1,\alpha +1,\frac {1-x}{2}\right ) \]