Internal problem ID [4521]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 2\).
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 54
Order:=6; dsolve(x^2*diff(y(x),x$2)-diff(y(x),x)+y(x)=0,y(x),type='series',x=2);
\[ y \relax (x ) = \left (1-\frac {\left (x -2\right )^{2}}{8}+\frac {\left (x -2\right )^{3}}{32}-\frac {3 \left (x -2\right )^{4}}{512}+\frac {\left (x -2\right )^{5}}{2048}\right ) y \relax (2)+\left (x -2+\frac {\left (x -2\right )^{2}}{8}-\frac {7 \left (x -2\right )^{3}}{96}+\frac {37 \left (x -2\right )^{4}}{1536}-\frac {211 \left (x -2\right )^{5}}{30720}\right ) D\relax (y )\relax (2)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 87
AsymptoticDSolveValue[x^2*y''[x]-y'[x]+y[x]==0,y[x],{x,2,5}]
\[ y(x)\to c_1 \left (\frac {(x-2)^5}{2048}-\frac {3}{512} (x-2)^4+\frac {1}{32} (x-2)^3-\frac {1}{8} (x-2)^2+1\right )+c_2 \left (-\frac {211 (x-2)^5}{30720}+\frac {37 (x-2)^4}{1536}-\frac {7}{96} (x-2)^3+\frac {1}{8} (x-2)^2+x-2\right ) \]