61.30.3 problem 151

Internal problem ID [12572]
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 : 151
Date solved : Monday, March 31, 2025 at 05:39:38 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 50

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

\[ y(x)\to c_1 \cos \left (\sqrt {a} \log \left (\sqrt {x^2-1}+x\right )\right )+c_2 \sin \left (\sqrt {a} \log \left (\sqrt {x^2-1}+x\right )\right ) \]

✗ Sympy

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

False