61.32.16 problem 225

Internal problem ID [12647]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 225
Date solved : Monday, March 31, 2025 at 06:49:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 20

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

\[ y(x)\to c_1 P_{\nu }^n(x)+c_2 Q_{\nu }^n(x) \]

✗ Sympy

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

False