59.2.6 problem 6

Internal problem ID [9999]
Book : Collection of Kovacic problems
Section : section 2. Solution found using all possible Kovacic cases
Problem number : 6
Date solved : Wednesday, March 05, 2025 at 08:02:00 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

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

\[ y = \sqrt {x}\, \left (c_{1} \sin \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )+c_{2} \cos \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 42

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

\[ y(x)\to \sqrt {x} \left (c_1 \cos \left (\frac {1}{2} \sqrt {3} \log (x)\right )+c_2 \sin \left (\frac {1}{2} \sqrt {3} \log (x)\right )\right ) \]

✓ Sympy. Time used: 0.088 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} \sin {\left (\frac {\sqrt {3} \log {\left (x \right )}}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} \log {\left (x \right )}}{2} \right )}\right ) \]