42.1.4 problem 3.6 (c)

Internal problem ID [6826]
Book : Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section : Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number : 3.6 (c)
Date solved : Sunday, March 30, 2025 at 11:23:50 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*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 13

Order:=6; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+12*y(x) = 0; 
ic:=y(0) = 0, D(y)(0) = 3; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = -5 x^{3}+3 x \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 12

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+12*y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==3}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to 3 x-5 x^3 \]

✓ Sympy. Time used: 0.965 (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 = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 3} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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