Internal problem ID [4509]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number: 17.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Lienard]
Solve \begin {gather*} \boxed {w^{\prime \prime }-x^{2} w^{\prime }+w=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 44
Order:=6; dsolve(diff(w(x),x$2)-x^2*diff(w(x),x)+w(x)=0,w(x),type='series',x=0);
\[ w \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{20} x^{5}\right ) w \relax (0)+\left (x -\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{120} x^{5}\right ) D\relax (w )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 56
AsymptoticDSolveValue[w''[x]-x^2*w'[x]+w[x]==0,w[x],{x,0,5}]
\[ w(x)\to c_2 \left (\frac {x^5}{120}+\frac {x^4}{12}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {x^5}{20}+\frac {x^4}{24}-\frac {x^2}{2}+1\right ) \]