Internal problem ID [4520]
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: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-x y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 2\).
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 49
Order:=6; dsolve(x^2*diff(y(x),x$2)-x*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=2);
\[ y \relax (x ) = \left (1-\frac {\left (x -2\right )^{2}}{4}+\frac {\left (x -2\right )^{3}}{24}-\frac {\left (x -2\right )^{4}}{192}\right ) y \relax (2)+\left (x -2+\frac {\left (x -2\right )^{2}}{4}-\frac {\left (x -2\right )^{3}}{12}+\frac {\left (x -2\right )^{4}}{48}-\frac {\left (x -2\right )^{5}}{192}\right ) D\relax (y )\relax (2)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 78
AsymptoticDSolveValue[x^2*y''[x]-x*y'[x]+2*y[x]==0,y[x],{x,2,5}]
\[ y(x)\to c_1 \left (-\frac {1}{192} (x-2)^4+\frac {1}{24} (x-2)^3-\frac {1}{4} (x-2)^2+1\right )+c_2 \left (-\frac {1}{192} (x-2)^5+\frac {1}{48} (x-2)^4-\frac {1}{12} (x-2)^3+\frac {1}{4} (x-2)^2+x-2\right ) \]