44.20.4 problem 3(b)

Internal problem ID [9418]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.5. More on Regular Singular Points. Page 183
Problem number : 3(b)
Date solved : Tuesday, September 30, 2025 at 06:18:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 55

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

\[ y = c_1 \,x^{2} \left (1+\frac {1}{2} x +\frac {1}{20} x^{2}-\frac {1}{60} x^{3}-\frac {1}{210} x^{4}-\frac {1}{3360} x^{5}+\frac {1}{20160} x^{6}+\frac {1}{100800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (12+6 x +6 x^{2}+5 x^{3}+x^{4}-\frac {1}{5} x^{5}-\frac {1}{10} x^{6}-\frac {3}{280} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 96

ode=x^2*D[y[x],{x,2}]-x^2*D[y[x],x]+(x^2-2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{120}-\frac {x^4}{60}+\frac {x^3}{12}+\frac {5 x^2}{12}+\frac {x}{2}+\frac {1}{x}+\frac {1}{2}\right )+c_2 \left (\frac {x^8}{20160}-\frac {x^7}{3360}-\frac {x^6}{210}-\frac {x^5}{60}+\frac {x^4}{20}+\frac {x^3}{2}+x^2\right ) \]

✓ Sympy. Time used: 0.365 (sec). Leaf size: 78

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

\[ y{\left (x \right )} = C_{2} x^{2} \left (- \frac {x^{5}}{3360} - \frac {x^{4}}{210} - \frac {x^{3}}{60} + \frac {x^{2}}{20} + \frac {x}{2} + 1\right ) + \frac {C_{1} \left (\frac {x^{8}}{5040} + \frac {11 x^{7}}{10080} - \frac {x^{6}}{720} - \frac {3 x^{5}}{80} - \frac {x^{4}}{8} + \frac {x^{2}}{2} + \frac {x}{2} + 1\right )}{x} + O\left (x^{8}\right ) \]