2.31.12 Problem 160

Internal problem ID [13821]
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 : Friday, December 19, 2025 at 01:36:09 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*}

2.31.12.1 ✓ Maple. Time used: 0.013 (sec). Leaf size: 61

ode:=(-x^2+1)*diff(diff(y(x),x),x)+(beta-alpha-(alpha+beta+2)*x)*diff(y(x),x)+n*(n+alpha+beta+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying a Liouvillian solution using Kovacics algorithm 
<- No Liouvillian solutions exists 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Kummer 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
   -> hypergeometric 
      -> heuristic approach 
      <- heuristic approach successful 
   <- hypergeometric successful 
<- special function solution successful
 

2.31.12.2 ✓ Mathematica. Time used: 0.129 (sec). Leaf size: 69

ode=(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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

2.31.12.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
n = symbols("n") 
y = Function("y") 
ode = Eq(n*(Alpha + BETA + n + 1)*y(x) + (1 - x**2)*Derivative(y(x), (x, 2)) + (-Alpha + BETA - x*(Alpha + BETA + 2))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False