55.31.5 problem 153

Internal problem ID [13926]
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 : 153
Date solved : Friday, October 03, 2025 at 06:55:48 AM
CAS classification : [_Gegenbauer]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 15

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

\[ y = c_1 \operatorname {LegendreP}\left (n , x\right )+c_2 \operatorname {LegendreQ}\left (n , x\right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 18

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

\begin{align*} y(x)&\to c_1 \operatorname {LegendreP}(n,x)+c_2 \operatorname {LegendreQ}(n,x) \end{align*}

✗ Sympy

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

False