23.3.362 problem 366

Internal problem ID [6076]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 366
Date solved : Friday, October 03, 2025 at 01:46:08 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: 0.014 (sec). Leaf size: 15

ode:=n*(n+1)*y(x)-2*x*diff(y(x),x)+(-x^2+1)*diff(diff(y(x),x),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: 0.015 (sec). Leaf size: 18

ode=n*(1 + n)*y[x] - 2*x*D[y[x],x] + (1 - x^2)*D[y[x],{x,2}] == 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