58.2.17 problem 18

Internal problem ID [9140]
Book : Second order enumerated odes
Section : section 2
Problem number : 18
Date solved : Sunday, March 30, 2025 at 02:23:15 PM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 42

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

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

✗ Sympy

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

False