9.8.9 problem problem 9

Internal problem ID [1074]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.2 Power series solutions. Page 624
Problem number : problem 9
Date solved : Saturday, March 29, 2025 at 10:38:17 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }+8 x y^{\prime }+12 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 39

Order:=6; 
ode:=(x^2-1)*diff(diff(y(x),x),x)+8*x*diff(y(x),x)+12*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (15 x^{4}+6 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {10}{3} x^{3}+7 x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 36

ode=(x^2-1)*D[y[x],{x,2}]+8*x*D[y[x],x]+12*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (7 x^5+\frac {10 x^3}{3}+x\right )+c_1 \left (15 x^4+6 x^2+1\right ) \]

✓ Sympy. Time used: 0.834 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x*Derivative(y(x), x) + (x**2 - 1)*Derivative(y(x), (x, 2)) + 12*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (15 x^{4} + 6 x^{2} + 1\right ) + C_{1} x \left (\frac {10 x^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]