45.3.6 problem 6

Internal problem ID [7264]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.3.1 page 250
Problem number : 6
Date solved : Sunday, March 30, 2025 at 11:52:49 AM
CAS classification : [_Bessel]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 44

Order:=6; 
ode:=diff(y(x),x)+x*diff(diff(y(x),x),x)+(x-4/x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{4} \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 52

ode=D[x*D[y[x],x],x]+(x-4/x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {\left (x^2+8\right )^2}{64 x^2}-\frac {1}{16} x^2 \log (x)\right )+c_2 \left (\frac {x^6}{384}-\frac {x^4}{12}+x^2\right ) \]

✓ Sympy. Time used: 0.963 (sec). Leaf size: 17

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,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{2} \left (1 - \frac {x^{2}}{12}\right ) + O\left (x^{6}\right ) \]