67.26.1 problem 36.2 (a)

Internal problem ID [17028]
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 (a)
Date solved : Thursday, October 02, 2025 at 01:42:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x 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: 0.020 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 44

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

\[ y(x)\to c_1 \left (\frac {x^5}{24}+\frac {x^3}{2}+x\right )+c_2 \left (\frac {x^6}{120}+\frac {x^4}{6}+x^2\right ) \]

✓ Sympy. Time used: 0.328 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{2} x^{2} \left (\frac {x^{2}}{6} + 1\right ) + C_{1} x \left (\frac {x^{4}}{24} + \frac {x^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]