67.23.18 problem 33.5 (f)

Internal problem ID [16951]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.5 (f)
Date solved : Thursday, October 02, 2025 at 01:40:37 PM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 34

Order:=6; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (-2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}-\frac {1}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 33

ode=(1-x^2)*D[y[x],{x,2}]-x*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (1-2 x^2\right )+c_2 \left (-\frac {x^5}{8}-\frac {x^3}{2}+x\right ) \]

✓ Sympy. Time used: 0.256 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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