59.1.428 problem 440

Internal problem ID [9600]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 440
Date solved : Wednesday, March 05, 2025 at 07:51:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 21

ode:=4*x^2*diff(diff(y(x),x),x)-4*x*diff(y(x),x)+(-16*x^2+3)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x}\, \left (c_{1} \sinh \left (2 x \right )+c_{2} \cosh \left (2 x \right )\right ) \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 32

ode=4*x^2*D[y[x],{x,2}]-4*x*D[y[x],x]+(3-16*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.222 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) - 4*x*Derivative(y(x), x) + (3 - 16*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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