Internal problem ID [11898]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 6, Series solutions of linear differential equations. Section 6.1. Exercises page
232
Problem number: 15.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 49
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+y(x)=0,y(x),type='series',x=1);
\[ y \left (x \right ) = \left (1-\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{2}-\frac {5 \left (-1+x \right )^{4}}{12}+\frac {\left (-1+x \right )^{5}}{3}\right ) y \left (1\right )+\left (-1+x -\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{6}-\frac {\left (-1+x \right )^{5}}{12}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 78
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+y[x]==0,y[x],{x,1,5}]
\[ y(x)\to c_1 \left (\frac {1}{3} (x-1)^5-\frac {5}{12} (x-1)^4+\frac {1}{2} (x-1)^3-\frac {1}{2} (x-1)^2+1\right )+c_2 \left (-\frac {1}{12} (x-1)^5+\frac {1}{6} (x-1)^3-\frac {1}{2} (x-1)^2+x-1\right ) \]