67.25.21 problem 35.4 (g)

Internal problem ID [17016]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.4 (g)
Date solved : Thursday, October 02, 2025 at 01:41:55 PM
CAS classification : [_Bessel]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{x^{2}}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 46

Order:=6; 
ode:=diff(diff(y(x),x),x)+diff(y(x),x)/x+(1-1/x^2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 58

ode=D[y[x],{x,2}]+1/x*D[y[x],x]+(1-1/x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{192}-\frac {x^3}{8}+x\right )+c_1 \left (\frac {1}{16} x \left (x^2-8\right ) \log (x)-\frac {5 x^4-16 x^2-64}{64 x}\right ) \]

✓ Sympy. Time used: 0.346 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - 1/x**2)*y(x) + Derivative(y(x), (x, 2)) + Derivative(y(x), x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{192} - \frac {x^{2}}{8} + 1\right ) + O\left (x^{6}\right ) \]