52.3.6 problem 6

Internal problem ID [8292]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.4 SPECIAL FUNCTIONS. EXERCISES 6.4. Page 267
Problem number : 6
Date solved : Wednesday, March 05, 2025 at 05:34:13 AM
CAS classification : [_Bessel]

\begin{align*} y^{\prime }+x y^{\prime \prime }+\left (x -\frac {4}{x}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 18

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

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

✓ Sympy. Time used: 0.210 (sec). Leaf size: 12

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

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