Internal problem ID [13672]
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: 35.4 (f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Lienard]
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 41
Order:=6; dsolve(diff(y(x),x$2)+1/x*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 60
AsymptoticDSolveValue[y''[x]+1/x*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (-\frac {3 x^4}{128}+\frac {x^2}{4}+\left (\frac {x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]