54.3.5 problem 5

Internal problem ID [8561]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 5
Date solved : Monday, January 27, 2025 at 04:16:25 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-4 x^{2}+1\right ) y^{\prime \prime }+8 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 39

Order:=8; 
dsolve((1-4*x^2)*diff(y(x),x$2)+8*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (-4 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {4}{3} x^{3}-\frac {16}{15} x^{5}-\frac {64}{35} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 40

AsymptoticDSolveValue[(1-4*x^2)*D[y[x],{x,2}]+8*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (1-4 x^2\right )+c_2 \left (-\frac {64 x^7}{35}-\frac {16 x^5}{15}-\frac {4 x^3}{3}+x\right ) \]