58.15.21 problem 21

Internal problem ID [14879]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 21
Date solved : Thursday, October 02, 2025 at 09:56:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+8 \left (x^{2}-1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 35

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+8*(x^2-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{4} \left (1-\frac {1}{2} x^{2}+\frac {1}{10} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-86400-86400 x^{2}-86400 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 36

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]+8*(x^2-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (x^2+\frac {1}{x^2}+1\right )+c_2 \left (\frac {x^8}{10}-\frac {x^6}{2}+x^4\right ) \]

✓ Sympy. Time used: 0.273 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = C_{1} x^{4} + O\left (x^{6}\right ) \]