59.1.353 problem 360

Internal problem ID [9525]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 360
Date solved : Sunday, March 30, 2025 at 02:36:48 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 21

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

\[ y = \frac {c_1 \sin \left (x^{2}\right )+c_2 \cos \left (x^{2}\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 41

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

\[ y(x)\to \frac {4 c_1 e^{-i x^2}-i c_2 e^{i x^2}}{4 x^2} \]

✓ Sympy. Time used: 0.193 (sec). Leaf size: 20

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

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