59.1.328 problem 335

Internal problem ID [9500]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 335
Date solved : Sunday, March 30, 2025 at 02:36:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 21

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

\[ y(x)\to x (c_1 j_1(x)-c_2 y_1(x)) \]

✓ Sympy. Time used: 0.085 (sec). Leaf size: 20

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

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