55.31.7 problem 155

Internal problem ID [13928]
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 : 155
Date solved : Thursday, October 02, 2025 at 08:09:12 AM
CAS classification : [_Gegenbauer]

\begin{align*} n \left (n +2\right ) y-3 x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 68

ode:=(-x^2+1)*diff(diff(y(x),x),x)-3*x*diff(y(x),x)+n*(n+2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 42

ode=(1-x^2)*D[y[x],{x,2}]-3*x*D[y[x],x]+n*(n+2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1 P_{n+\frac {1}{2}}^{\frac {1}{2}}(x)+c_2 Q_{n+\frac {1}{2}}^{\frac {1}{2}}(x)}{\sqrt [4]{x^2-1}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(n*(n + 2)*y(x) - 3*x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False