Internal problem ID [13986]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {\left (x^{2}+4\right )^{2} y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 34
Order:=6; dsolve((4+x^2)^2*diff(y(x),x$2)+y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (1-\frac {1}{32} x^{2}+\frac {17}{6144} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{96} x^{3}+\frac {49}{30720} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 42
AsymptoticDSolveValue[(4+x^2)^2*y''[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {49 x^5}{30720}-\frac {x^3}{96}+x\right )+c_1 \left (\frac {17 x^4}{6144}-\frac {x^2}{32}+1\right ) \]