69.24.10 problem 750

Internal problem ID [18509]
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 : 750
Date solved : Thursday, October 02, 2025 at 03:14:29 PM
CAS classification : [_Lienard]

\begin{align*} y^{\prime \prime }+\frac {5 y^{\prime }}{x}+y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

ode:=diff(diff(y(x),x),x)+5*diff(y(x),x)/x+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\operatorname {BesselY}\left (0, x\right ) c_2 x -\operatorname {BesselJ}\left (0, x\right ) c_1 x +2 \operatorname {BesselY}\left (1, x\right ) c_2 +2 \operatorname {BesselJ}\left (1, x\right ) c_1}{x^{3}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 22

ode=D[y[x],{x,2}]+5/x*D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1 \operatorname {BesselJ}(2,x)+c_2 \operatorname {BesselY}(2,x)}{x^2} \end{align*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + Derivative(y(x), (x, 2)) + 5*Derivative(y(x), x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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