61.30.12 problem 160

Internal problem ID [12660]
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
Problem number : 160
Date solved : Tuesday, January 28, 2025 at 08:03:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \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 \end{align*}

✓ Solution by Maple

Time used: 0.279 (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)
 

\[ y = c_{1} \operatorname {hypergeom}\left (\left [-n , n +\alpha +\beta +1\right ], \left [\beta +1\right ], \frac {1}{2}+\frac {x}{2}\right )+c_{2} \left (\frac {1}{2}+\frac {x}{2}\right )^{-\beta } \operatorname {hypergeom}\left (\left [-n -\beta , n +\alpha +1\right ], \left [1-\beta \right ], \frac {1}{2}+\frac {x}{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.195 (sec). Leaf size: 69

DSolve[(1-x^2)*D[y[x],{x,2}]+(\[Beta]-\[Alpha]-(\[Alpha]+\[Beta]+2)*x)*D[y[x],x]+n*(n+\[Alpha]+\[Beta]+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 2^{\alpha } c_2 (x-1)^{-\alpha } \operatorname {Hypergeometric2F1}\left (-n-\alpha ,n+\beta +1,1-\alpha ,\frac {1-x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (-n,n+\alpha +\beta +1,\alpha +1,\frac {1-x}{2}\right ) \]