55.33.17 problem 226

Internal problem ID [14000]
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 : 226
Date solved : Friday, October 03, 2025 at 07:23:22 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 )-\mu ^{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)-mu^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 20

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

\begin{align*} y(x)&\to c_1 P_{\nu }^{\mu }(x)+c_2 Q_{\nu }^{\mu }(x) \end{align*}

✗ Sympy

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

False