59.2.4 problem 4

Internal problem ID [9997]
Book : Collection of Kovacic problems
Section : section 2. Solution found using all possible Kovacic cases
Problem number : 4
Date solved : Sunday, March 30, 2025 at 02:51:14 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }-\frac {y}{4 x^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 32

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

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

✓ Sympy. Time used: 0.073 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = \frac {C_{1}}{x^{- \frac {1}{2} + \frac {\sqrt {2}}{2}}} + C_{2} x^{\frac {1}{2} + \frac {\sqrt {2}}{2}} \]