10.13.8 problem 10

Internal problem ID [1368]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.2, Series Solutions Near an Ordinary Point, Part I. page 263
Problem number : 10
Date solved : Saturday, March 29, 2025 at 10:53:30 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-x^{2}+4\right ) y^{\prime \prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 34

Order:=6; 
ode:=(-x^2+4)*diff(diff(y(x),x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {x^{2}}{4}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{3}-\frac {1}{240} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 35

ode=(4-x^2)*D[y[x],{x,2}]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (1-\frac {x^2}{4}\right )+c_2 \left (-\frac {x^5}{240}-\frac {x^3}{12}+x\right ) \]

✓ Sympy. Time used: 0.746 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((4 - 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=6)
 

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {x^{2}}{4}\right ) + C_{1} x \left (1 - \frac {x^{2}}{12}\right ) + O\left (x^{6}\right ) \]