20.28.8 problem 8

Internal problem ID [4070]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Additional problems. Section 11.7. page 788
Problem number : 8
Date solved : Monday, January 27, 2025 at 08:07:29 AM
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: 0.002 (sec). Leaf size: 34

Order:=6; 
dsolve((1+4*x^2)*diff(y(x),x$2)-8*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \left (4 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {4}{3} x^{3}-\frac {16}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 33

AsymptoticDSolveValue[(1+4*x^2)*D[y[x],{x,2}]-8*y[x]==0,y[x],{x,0,"6"-1}]
 

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