53.1.382 problem 391

Internal problem ID [10854]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 391
Date solved : Tuesday, September 30, 2025 at 07:32:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 37

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

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

✓ Sympy. Time used: 0.132 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \frac {C_{1} J_{\frac {1}{2}}\left (x\right ) + C_{2} Y_{\frac {1}{2}}\left (x\right )}{x^{\frac {3}{2}}} \]