Internal problem ID [5860]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT
SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number: 30.
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: 71
Order:=8; dsolve(x*diff(y(x),x$2)+diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-x +\frac {1}{4} x^{2}-\frac {1}{36} x^{3}+\frac {1}{576} x^{4}-\frac {1}{14400} x^{5}+\frac {1}{518400} x^{6}-\frac {1}{25401600} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \left (\ln \relax (x ) c_{2}+c_{1}\right )+\left (2 x -\frac {3}{4} x^{2}+\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}+\frac {137}{432000} x^{5}-\frac {49}{5184000} x^{6}+\frac {121}{592704000} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 153
AsymptoticDSolveValue[x*y''[x]+y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {x^7}{25401600}+\frac {x^6}{518400}-\frac {x^5}{14400}+\frac {x^4}{576}-\frac {x^3}{36}+\frac {x^2}{4}-x+1\right )+c_2 \left (\frac {121 x^7}{592704000}-\frac {49 x^6}{5184000}+\frac {137 x^5}{432000}-\frac {25 x^4}{3456}+\frac {11 x^3}{108}-\frac {3 x^2}{4}+\left (-\frac {x^7}{25401600}+\frac {x^6}{518400}-\frac {x^5}{14400}+\frac {x^4}{576}-\frac {x^3}{36}+\frac {x^2}{4}-x+1\right ) \log (x)+2 x\right ) \]