61.30.13 problem 161

Internal problem ID [12661]
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 : 161
Date solved : Tuesday, January 28, 2025 at 08:03:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y&=0 \end{align*}

Solution by Maple

Time used: 0.303 (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)
 
\[ y = c_{1} \operatorname {hypergeom}\left (\left [n +1, -n -\alpha -\beta \right ], \left [1-\beta \right ], \frac {1}{2}+\frac {x}{2}\right )+c_{2} \left (\frac {1}{2}+\frac {x}{2}\right )^{\beta } \operatorname {hypergeom}\left (\left [-n -\alpha , n +\beta +1\right ], \left [\beta +1\right ], \frac {1}{2}+\frac {x}{2}\right ) \]

Solution by Mathematica

Time used: 0.160 (sec). Leaf size: 74

DSolve[(1-x^2)*D[y[x],{x,2}]+(\[Alpha]-\[Beta]+(\[Alpha]+\[Beta]-2)*x)*D[y[x],x]+(n+1)*(n+\[Alpha]+\[Beta])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 
\[ y(x)\to 2^{-\alpha } c_2 (x-1)^{\alpha } \operatorname {Hypergeometric2F1}\left (n+\alpha +1,-n-\beta ,\alpha +1,\frac {1-x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (n+1,-n-\alpha -\beta ,1-\alpha ,\frac {1-x}{2}\right ) \]