2.31.8 Problem 156

Internal problem ID [13817]
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 : 156
Date solved : Friday, December 19, 2025 at 01:20:26 PM
CAS classification : [_Gegenbauer]

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

2.31.8.1 ✓ Maple. Time used: 0.010 (sec). Leaf size: 27

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

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

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 
   <- Legendre successful 
<- special function solution successful
 

2.31.8.2 ✓ Mathematica. Time used: 0.028 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to \left (x^2-1\right )^{-n/2} (c_1 P_{\nu }^n(x)+c_2 Q_{\nu }^n(x)) \end{align*}

2.31.8.3 ✗ Sympy

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

False