20.24.12 problem Problem 12

Internal problem ID [3997]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page 739
Problem number : Problem 12
Date solved : Sunday, March 30, 2025 at 02:14:06 AM
CAS classification : [_Gegenbauer]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 25

ode=(x^2-1)*D[y[x],{x,2}]-6*x*D[y[x],x]+12*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (x^3+x\right )+c_1 \left (x^4+6 x^2+1\right ) \]

✓ Sympy. Time used: 0.743 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-6*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)
 

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