Internal problem ID [6202]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises
page 373
Problem number: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (x -2\right ) y^{\prime \prime }+2 \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: 0.063 (sec). Leaf size: 47
Order:=8; dsolve(x*(x-2)*diff(y(x),x$2)+2*(x-1)*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-x +\mathrm {O}\left (x^{8}\right )\right )+\left (\frac {5}{2} x -\frac {3}{8} x^{2}-\frac {1}{12} x^{3}-\frac {5}{192} x^{4}-\frac {3}{320} x^{5}-\frac {7}{1920} x^{6}-\frac {1}{672} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 71
AsymptoticDSolveValue[x*(x-2)*y''[x]+2*(x-1)*y'[x]-2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {x^7}{672}-\frac {7 x^6}{1920}-\frac {3 x^5}{320}-\frac {5 x^4}{192}-\frac {x^3}{12}-\frac {3 x^2}{8}+\frac {5 x}{2}+(1-x) \log (x)\right )+c_1 (1-x) \]