2.31.1 Problem 149

Internal problem ID [13810]
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 : 149
Date solved : Friday, December 19, 2025 at 01:04:19 PM
CAS classification : [_Gegenbauer]

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

2.31.1.1 ✓ Maple. Time used: 0.010 (sec). Leaf size: 52

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

\[ y = -\left (x -1\right ) \left (x +1\right ) \left (\operatorname {hypergeom}\left (\left [1+\frac {n}{2}, \frac {3}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) c_2 x +c_1 \operatorname {hypergeom}\left (\left [1-\frac {n}{2}, \frac {1}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )\right ) \]

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 
   -> Whittaker 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
   -> hypergeometric 
      -> heuristic approach 
      -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
      <- hyper3 successful: received ODE is equivalent to the 2F1 ODE 
   <- hypergeometric successful 
<- special function solution successful
 

2.31.1.2 ✓ Mathematica. Time used: 0.06 (sec). Leaf size: 56

ode=(1-x^2)*D[y[x],{x,2}]+n*(n-1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to i c_2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},\frac {n}{2},\frac {3}{2},x^2\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {n-1}{2},-\frac {n}{2},\frac {1}{2},x^2\right ) \end{align*}

2.31.1.3 ✓ Sympy. Time used: 0.209 (sec). Leaf size: 94

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

\[ y{\left (x \right )} = \frac {\left (C_{1} \sqrt {x^{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4} - \frac {\sqrt {\left (2 n - 1\right )^{2}}}{4}, \frac {\sqrt {\left (2 n - 1\right )^{2}}}{4} + \frac {1}{4} \\ \frac {3}{2} \end {matrix}\middle | {x^{2}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {\sqrt {\left (2 n - 1\right )^{2}}}{4} - \frac {1}{4}, \frac {\sqrt {\left (2 n - 1\right )^{2}}}{4} - \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{2}} \right )}\right ) \sqrt [4]{x^{2}}}{\sqrt {x}} \]