Internal problem ID [6274]
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: 26.
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: 0.063 (sec). Leaf size: 52
Order:=8; dsolve(x*diff(y(x),x$2)+(2-x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \left (1+\frac {1}{2} x +\frac {1}{6} x^{2}+\frac {1}{24} x^{3}+\frac {1}{120} x^{4}+\frac {1}{720} x^{5}+\frac {1}{5040} x^{6}+\frac {1}{40320} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.116 (sec). Leaf size: 90
AsymptoticDSolveValue[x*y''[x]+(2-x)*y'[x]-y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x^5}{720}+\frac {x^4}{120}+\frac {x^3}{24}+\frac {x^2}{6}+\frac {x}{2}+\frac {1}{x}+1\right )+c_2 \left (\frac {x^6}{5040}+\frac {x^5}{720}+\frac {x^4}{120}+\frac {x^3}{24}+\frac {x^2}{6}+\frac {x}{2}+1\right ) \]