Internal problem ID [6503]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 32.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 59
Order:=6; dsolve(x*diff(y(x), x$2) +diff(y(x),x)-y(x) = 0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {1}{576} x^{4}+\frac {1}{14400} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -\frac {3}{4} x^{2}-\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}-\frac {137}{432000} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 107
AsymptoticDSolveValue[x*y''[x] +y'[x]-y[x] == 0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^5}{14400}+\frac {x^4}{576}+\frac {x^3}{36}+\frac {x^2}{4}+x+1\right )+c_2 \left (-\frac {137 x^5}{432000}-\frac {25 x^4}{3456}-\frac {11 x^3}{108}-\frac {3 x^2}{4}+\left (\frac {x^5}{14400}+\frac {x^4}{576}+\frac {x^3}{36}+\frac {x^2}{4}+x+1\right ) \log (x)-2 x\right ) \]