49.20.1 problem 4

Internal problem ID [7730]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 182
Problem number : 4
Date solved : Wednesday, March 05, 2025 at 04:51:33 AM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

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

\[ y = \left (1-x^{2}-\frac {1}{3} x^{4}-\frac {1}{5} x^{6}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 32

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

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

✓ Sympy. Time used: 0.830 (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)) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{6}}{5} - \frac {x^{4}}{3} - x^{2} + 1\right ) + C_{1} x + O\left (x^{8}\right ) \]