Internal problem ID [11754]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 20, Series solutions of second order linear equations. Exercises page
195
Problem number: 20.2 (iii).
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: 0.016 (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: 0.004 (sec). Leaf size: 83
AsymptoticDSolveValue[2*x*y''[x]+y'[x]-2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {4 x^5}{155925}+\frac {2 x^4}{2835}+\frac {4 x^3}{315}+\frac {2 x^2}{15}+\frac {2 x}{3}+1\right )+c_2 \left (\frac {4 x^5}{14175}+\frac {2 x^4}{315}+\frac {4 x^3}{45}+\frac {2 x^2}{3}+2 x+1\right ) \]