53.1.329 problem 336

Internal problem ID [10801]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 336
Date solved : Tuesday, September 30, 2025 at 07:32:05 PM
CAS classification : [[_Emden, _Fowler]]

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

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

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

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

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 43

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

\begin{align*} y(x)&\to \frac {e^{-\frac {i x^2}{2}} \left (2 c_1-i c_2 e^{i x^2}\right )}{2 x^2} \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 24

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

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