68.16.24 problem 24 (b)

Internal problem ID [17817]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 24 (b)
Date solved : Thursday, October 02, 2025 at 02:28:31 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 }+9 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)+9*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {9}{2} x^{2}+\frac {15}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {4}{3} x^{3}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 35

ode=(1-x^2)*D[y[x],{x,2}]-x*D[y[x],x]+9*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (x-\frac {4 x^3}{3}\right )+c_1 \left (\frac {15 x^4}{8}-\frac {9 x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.261 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)) + 9*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {15 x^{4}}{8} - \frac {9 x^{2}}{2} + 1\right ) + C_{1} x \left (1 - \frac {4 x^{2}}{3}\right ) + O\left (x^{6}\right ) \]