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 : Monday, January 27, 2025 at 04:56:14 AM
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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[(4-x^2)*D[y[x],{x,2}]+2*y[x]==0,y[x],{x,0,"6"-1}]
 

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