67.26.2 problem 36.2 (b)

Internal problem ID [17029]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page 739
Problem number : 36.2 (b)
Date solved : Thursday, October 02, 2025 at 01:42:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 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: 0.029 (sec). Leaf size: 47

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

\[ y = c_1 \,x^{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12+12 x +6 x^{2}+4 x^{3}+\frac {5}{2} x^{4}+\frac {11}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 62

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

\[ y(x)\to c_1 \left (\frac {5 x^3}{24}+\frac {x^2}{3}+\frac {x}{2}+\frac {1}{x}+1\right )+c_2 \left (\frac {x^6}{24}+\frac {x^5}{6}+\frac {x^4}{2}+x^3+x^2\right ) \]

✓ Sympy. Time used: 0.310 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*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=6)
 

\[ y{\left (x \right )} = C_{2} x^{2} \left (\frac {x^{3}}{6} + \frac {x^{2}}{2} + x + 1\right ) + \frac {C_{1} \left (- \frac {19 x^{6}}{720} - \frac {3 x^{5}}{40} - \frac {x^{4}}{8} + \frac {x^{2}}{2} + x + 1\right )}{x} + O\left (x^{6}\right ) \]