7.15.11 problem 11

Internal problem ID [467]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 11
Date solved : Tuesday, March 04, 2025 at 11:24:49 AM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 31

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

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

✓ Sympy. Time used: 0.800 (sec). Leaf size: 31

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)) + 12*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (3 x^{4} - 6 x^{2} + 1\right ) + C_{1} x \left (1 - \frac {5 x^{2}}{3}\right ) + O\left (x^{6}\right ) \]