85.72.13 problem 2 (c)

Internal problem ID [22949]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. A Exercises at page 316
Problem number : 2 (c)
Date solved : Thursday, October 02, 2025 at 09:16:52 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+6 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)-2*x*diff(y(x),x)+6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

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

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.294 (sec). Leaf size: 26

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

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