59.3.4 problem Kovacic 1985 paper. page 19. section 4.2. Example 1

Internal problem ID [10006]
Book : Collection of Kovacic problems
Section : section 3. Problems from Kovacic related papers
Problem number : Kovacic 1985 paper. page 19. section 4.2. Example 1
Date solved : Sunday, March 30, 2025 at 02:51:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\left (\frac {1}{x}-\frac {3}{16 x^{2}}\right ) y \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x) = (1/x-3/16/x^2)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{{1}/{4}} \left (c_1 \sinh \left (2 \sqrt {x}\right )+c_2 \cosh \left (2 \sqrt {x}\right )\right ) \]

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 41

ode=D[y[x],{x,2}]== (1/x-3/(16*x^2))*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} e^{-2 \sqrt {x}} \sqrt [4]{x} \left (2 c_1 e^{4 \sqrt {x}}-c_2\right ) \]

✓ Sympy. Time used: 0.101 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(1/x - 3/(16*x**2))*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {1}{2}}\left (2 i \sqrt {x}\right ) + C_{2} Y_{\frac {1}{2}}\left (2 i \sqrt {x}\right )\right ) \]