1.17.4 problem 4

Internal problem ID [517]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.6 (Applications of Bessel functions). Problems at page 261
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 03:59:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-5 x y^{\prime }+\left (8+x \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 55

ode:=x^2*diff(diff(y(x),x),x)-5*x*diff(y(x),x)+(8+x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\operatorname {BesselY}\left (0, 2 \sqrt {x}\right ) x^{3} c_2 +\operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) x^{{5}/{2}} c_2 +\operatorname {BesselJ}\left (1, 2 \sqrt {x}\right ) x^{{5}/{2}} c_1 -\operatorname {BesselJ}\left (0, 2 \sqrt {x}\right ) x^{3} c_1 \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 36

ode=x^2*D[y[x],{x,2}]-5*x*D[y[x],x]+(8+x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 2 x^3 \left (c_1 \operatorname {BesselJ}\left (2,2 \sqrt {x}\right )-c_2 \operatorname {BesselY}\left (2,2 \sqrt {x}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 5*x*Derivative(y(x), x) + (x + 8)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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