47.4.5 problem 53

Internal problem ID [7482]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.2 problems. page 95
Problem number : 53
Date solved : Sunday, March 30, 2025 at 12:10:17 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{x}+x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+1/x*diff(y(x),x)+x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {BesselJ}\left (0, \frac {x^{2}}{2}\right )+c_2 \operatorname {BesselY}\left (0, \frac {x^{2}}{2}\right ) \]

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 31

ode=D[y[x],{x,2}]+1/x*D[y[x],x]+x^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \operatorname {BesselJ}\left (0,\frac {x^2}{2}\right )+2 c_2 \operatorname {BesselY}\left (0,\frac {x^2}{2}\right ) \]

✓ Sympy. Time used: 0.195 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = C_{1} J_{0}\left (\frac {x^{2}}{2}\right ) + C_{2} Y_{0}\left (\frac {x^{2}}{2}\right ) \]