Internal problem ID [6251]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-\left (2+x \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: 0.025 (sec). Leaf size: 70
Order:=8; dsolve(x*diff(y(x),x$2)-(2+x)*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1+\frac {5}{4} x +\frac {3}{4} x^{2}+\frac {7}{24} x^{3}+\frac {1}{12} x^{4}+\frac {3}{160} x^{5}+\frac {1}{288} x^{6}+\frac {11}{20160} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{1} x^{3}+c_{2} \left (\left (24 x^{3}+30 x^{4}+18 x^{5}+7 x^{6}+2 x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \ln \relax (x )+\left (12-12 x +18 x^{2}+26 x^{3}+x^{4}-9 x^{5}-6 x^{6}-\frac {9}{4} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.077 (sec). Leaf size: 115
AsymptoticDSolveValue[x*y''[x]-(2+x)*y'[x]-2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {1}{12} \left (7 x^3+18 x^2+30 x+24\right ) x^3 \log (x)+\frac {1}{36} \left (-25 x^6-45 x^5-27 x^4+54 x^3+54 x^2-36 x+36\right )\right )+c_2 \left (\frac {x^9}{288}+\frac {3 x^8}{160}+\frac {x^7}{12}+\frac {7 x^6}{24}+\frac {3 x^5}{4}+\frac {5 x^4}{4}+x^3\right ) \]