85.77.3 problem 5 (b)

Internal problem ID [22986]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. A Exercises at page 341
Problem number : 5 (b)
Date solved : Thursday, October 02, 2025 at 09:17:15 PM
CAS classification : [_Bessel]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 77

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^2-8)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{-2 \sqrt {2}} \left (1+\frac {1}{-4+8 \sqrt {2}} x^{2}+\frac {1}{320-192 \sqrt {2}} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{2 \sqrt {2}} \left (1-\frac {1}{4+8 \sqrt {2}} x^{2}+\frac {1}{320+192 \sqrt {2}} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 230

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

\[ y(x)\to c_1 \left (\frac {x^4}{\left (-6+2 \sqrt {2}+\left (1+2 \sqrt {2}\right ) \left (2+2 \sqrt {2}\right )\right ) \left (-4+2 \sqrt {2}+\left (3+2 \sqrt {2}\right ) \left (4+2 \sqrt {2}\right )\right )}-\frac {x^2}{-6+2 \sqrt {2}+\left (1+2 \sqrt {2}\right ) \left (2+2 \sqrt {2}\right )}+1\right ) x^{2 \sqrt {2}}+c_2 \left (\frac {x^4}{\left (-6-2 \sqrt {2}+\left (1-2 \sqrt {2}\right ) \left (2-2 \sqrt {2}\right )\right ) \left (-4-2 \sqrt {2}+\left (3-2 \sqrt {2}\right ) \left (4-2 \sqrt {2}\right )\right )}-\frac {x^2}{-6-2 \sqrt {2}+\left (1-2 \sqrt {2}\right ) \left (2-2 \sqrt {2}\right )}+1\right ) x^{-2 \sqrt {2}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x**2 - 8)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

NotImplementedError : Not sure of sign of 6 - x0