Internal problem ID [10995]
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: 161.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (-2+\beta +\alpha \right ) x \right ) y^{\prime }+\left (1+n \right ) \left (n +\alpha +\beta \right ) y=0} \]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 64
dsolve((1-x^2)*diff(y(x),x$2)+(alpha-beta+(alpha+beta-2)*x)*diff(y(x),x)+(n+1)*(n+alpha+beta)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [n +1, -n -\alpha -\beta \right ], \left [1-\beta \right ], \frac {x}{2}+\frac {1}{2}\right )+c_{2} \left (\frac {x}{2}+\frac {1}{2}\right )^{\beta } \operatorname {hypergeom}\left (\left [-n -\alpha , n +\beta +1\right ], \left [\beta +1\right ], \frac {x}{2}+\frac {1}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.225 (sec). Leaf size: 74
DSolve[(1-x^2)*y''[x]+(\[Alpha]-\[Beta]+(\[Alpha]+\[Beta]-2)*x)*y'[x]+(n+1)*(n+\[Alpha]+\[Beta])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{-\alpha } c_2 (x-1)^{\alpha } \operatorname {Hypergeometric2F1}\left (n+\alpha +1,-n-\beta ,\alpha +1,\frac {1-x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (n+1,-n-\alpha -\beta ,1-\alpha ,\frac {1-x}{2}\right ) \]