73.25.14 problem 35.3 (h)

Internal problem ID [15722]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.3 (h)
Date solved : Tuesday, January 28, 2025 at 08:06:00 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 39

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

\[ y = \left (1-\frac {1}{32} x^{2}+\frac {17}{6144} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{96} x^{3}+\frac {49}{30720} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (\frac {49 x^5}{30720}-\frac {x^3}{96}+x\right )+c_1 \left (\frac {17 x^4}{6144}-\frac {x^2}{32}+1\right ) \]