59.1.55 problem 57

Internal problem ID [9227]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 57
Date solved : Sunday, March 30, 2025 at 02:26:02 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }-4 y&=0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 39

ode:=(-x^2+1)*diff(diff(y(x),x),x)-5*x*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\ln \left (x +\sqrt {x^{2}-1}\right ) c_2 x -\sqrt {x^{2}-1}\, c_2 +c_1 x}{\left (x^{2}-1\right )^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.155 (sec). Leaf size: 47

ode=(1-x^2)*D[y[x],{x,2}]-5*x*D[y[x],x]-4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {c_2 x \arcsin (x)}{\left (1-x^2\right )^{3/2}}+\frac {c_1 x}{\left (x^2-1\right )^{3/2}}-\frac {c_2}{x^2-1} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-5*x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)) - 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False