Internal problem ID [6250]
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: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-\left (2+x \right ) y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.028 (sec). Leaf size: 68
Order:=8; dsolve(x*diff(y(x),x$2)-(2+x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{3} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \relax (x ) \left (6 x^{3}+6 x^{4}+3 x^{5}+x^{6}+\frac {1}{4} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (12-6 x +6 x^{2}+11 x^{3}+5 x^{4}+x^{5}-\frac {1}{16} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.071 (sec). Leaf size: 104
AsymptoticDSolveValue[x*y''[x]-(2+x)*y'[x]-y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {1}{12} \left (x^3+3 x^2+6 x+6\right ) x^3 \log (x)+\frac {1}{36} \left (-x^6+9 x^4+27 x^3+18 x^2-18 x+36\right )\right )+c_2 \left (\frac {x^9}{720}+\frac {x^8}{120}+\frac {x^7}{24}+\frac {x^6}{6}+\frac {x^5}{2}+x^4+x^3\right ) \]