Internal problem ID [5736]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Problems for review and discovert.
(A) Drill Exercises . Page 194
Problem number: 2(f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-\left (x -1\right ) y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 51
Order:=8; dsolve(x*diff(y(x),x$2)-(x-1)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (5 x -\frac {9}{4} x^{2}+\frac {1}{18} x^{3}+\frac {1}{288} x^{4}+\frac {1}{3600} x^{5}+\frac {1}{43200} x^{6}+\frac {1}{529200} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}+\left (1-2 x +\frac {1}{2} x^{2}+\mathrm {O}\left (x^{8}\right )\right ) \left (\ln \relax (x ) c_{2}+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 83
AsymptoticDSolveValue[x*y''[x]-(x-1)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x^2}{2}-2 x+1\right )+c_2 \left (\frac {x^7}{529200}+\frac {x^6}{43200}+\frac {x^5}{3600}+\frac {x^4}{288}+\frac {x^3}{18}-\frac {9 x^2}{4}+\left (\frac {x^2}{2}-2 x+1\right ) \log (x)+5 x\right ) \]