48.3.3 problem Example 3.32

Internal problem ID [7527]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.5 HIGHER ORDER ODE. Page 181
Problem number : Example 3.32
Date solved : Sunday, March 30, 2025 at 12:13:42 PM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }-\frac {x y^{\prime }}{-x^{2}+1}+\frac {y}{-x^{2}+1}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

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

\[ y = c_1 x +c_2 \sqrt {x -1}\, \sqrt {x +1} \]

✓ Mathematica. Time used: 0.093 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \cosh \left (\frac {\sqrt {1-x^2} \arcsin (x)}{\sqrt {x^2-1}}\right )+i c_2 \sinh \left (\frac {\sqrt {1-x^2} \arcsin (x)}{\sqrt {x^2-1}}\right ) \]

✗ Sympy

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

False