Internal problem ID [4519]
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: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}-2 x \right ) y^{\prime \prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 32
Order:=6; dsolve((x^2-2*x)*diff(y(x),x$2)+2*y(x)=0,y(x),type='series',x=1);
\[ y \relax (x ) = \left (1+\left (x -1\right )^{2}+\frac {\left (x -1\right )^{4}}{3}\right ) y \relax (1)+\left (x -1+\frac {\left (x -1\right )^{3}}{3}+\frac {2 \left (x -1\right )^{5}}{15}\right ) D\relax (y )\relax (1)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 47
AsymptoticDSolveValue[(x^2-2*x)*y''[x]+2*y[x]==0,y[x],{x,1,5}]
\[ y(x)\to c_1 \left (\frac {1}{3} (x-1)^4+(x-1)^2+1\right )+c_2 \left (\frac {2}{15} (x-1)^5+\frac {1}{3} (x-1)^3+x-1\right ) \]