Internal problem ID [5857]
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: 27.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-x y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 46
Order:=8; dsolve(x*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \ln \relax (x ) \left (-x +\mathrm {O}\left (x^{8}\right )\right ) c_{2}+c_{1} x \left (1+\mathrm {O}\left (x^{8}\right )\right )+\left (1+x -\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{72} x^{4}-\frac {1}{480} x^{5}-\frac {1}{3600} x^{6}-\frac {1}{30240} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.139 (sec). Leaf size: 51
AsymptoticDSolveValue[x*y''[x]-x*y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {-2 x^6-15 x^5-100 x^4-600 x^3-3600 x^2+14400 x+7200}{7200}-x \log (x)\right )+c_2 x \]