Internal problem ID [4757]
Book: Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section: Chapter 7. POWER SERIES METHODS. 7.2.1 Exercises. page 290
Problem number: 7.2.8 part(a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y=0} \end {gather*} 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((x^2+1)*diff(y(x),x$2)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \relax (0)+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 42
AsymptoticDSolveValue[(x^2+1)*y''[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {7 x^5}{120}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^4}{8}-\frac {x^2}{2}+1\right ) \]