Internal problem ID [11911]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius).
Exercises page 251
Problem number: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {2 x y^{\prime \prime }+y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 44
Order:=6; dsolve(2*x*diff(y(x),x$2)+diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {2}{3} x +\frac {2}{15} x^{2}-\frac {4}{315} x^{3}+\frac {2}{2835} x^{4}-\frac {4}{155925} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x +\frac {2}{3} x^{2}-\frac {4}{45} x^{3}+\frac {2}{315} x^{4}-\frac {4}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 83
AsymptoticDSolveValue[2*x*y''[x]+y'[x]+2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {4 x^5}{155925}+\frac {2 x^4}{2835}-\frac {4 x^3}{315}+\frac {2 x^2}{15}-\frac {2 x}{3}+1\right )+c_2 \left (-\frac {4 x^5}{14175}+\frac {2 x^4}{315}-\frac {4 x^3}{45}+\frac {2 x^2}{3}-2 x+1\right ) \]