54.3.228 problem 1244

Internal problem ID [12523]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1244
Date solved : Friday, October 03, 2025 at 03:30:41 AM
CAS classification : [_Gegenbauer]

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 32

ode=(n - v)*(1 + n + v)*y[x] + 2*(1 + n)*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 \left (x^2-1\right )^{-n/2} (c_1 P_v^n(x)+c_2 Q_v^n(x)) \end{align*}

✗ Sympy

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

False