75.24.7 problem 747

Internal problem ID [17148]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.2. Expanding a solution in generalized power series. Bessels equation. Exercises page 177
Problem number : 747
Date solved : Monday, March 31, 2025 at 03:43:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 22

ode=D[y[x],{x,2}]+1/x*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.182 (sec). Leaf size: 15

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

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