61.30.9 problem 157

Internal problem ID [12578]
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 : 157
Date solved : Monday, March 31, 2025 at 05:39:52 AM
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*}

✓ Maple. Time used: 0.026 (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);
 

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

✓ Mathematica. Time used: 0.044 (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]
 

\[ y(x)\to \left (x^2-1\right )^{n/2} (c_1 P_{\nu }^n(x)+c_2 Q_{\nu }^n(x)) \]

✗ 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