Internal problem ID [6171]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots
Nonintegral. Exercises page 365
Problem number: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime }+\left (-x +2\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: 0.022 (sec). Leaf size: 71
Order:=8; dsolve(2*x*diff(y(x),x$2)+(2-x)*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+x +\frac {3}{8} x^{2}+\frac {1}{12} x^{3}+\frac {5}{384} x^{4}+\frac {1}{640} x^{5}+\frac {7}{46080} x^{6}+\frac {1}{80640} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (-\frac {3}{2} x -\frac {13}{16} x^{2}-\frac {31}{144} x^{3}-\frac {173}{4608} x^{4}-\frac {187}{38400} x^{5}-\frac {463}{921600} x^{6}-\frac {971}{22579200} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 151
AsymptoticDSolveValue[2*x*y''[x]+(2-x)*y'[x]-2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x^7}{80640}+\frac {7 x^6}{46080}+\frac {x^5}{640}+\frac {5 x^4}{384}+\frac {x^3}{12}+\frac {3 x^2}{8}+x+1\right )+c_2 \left (-\frac {971 x^7}{22579200}-\frac {463 x^6}{921600}-\frac {187 x^5}{38400}-\frac {173 x^4}{4608}-\frac {31 x^3}{144}-\frac {13 x^2}{16}+\left (\frac {x^7}{80640}+\frac {7 x^6}{46080}+\frac {x^5}{640}+\frac {5 x^4}{384}+\frac {x^3}{12}+\frac {3 x^2}{8}+x+1\right ) \log (x)-\frac {3 x}{2}\right ) \]