Internal problem ID [6203]
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: 9.
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 \(x = 2\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 49
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=2);
\[ y \relax (x ) = \left (-\frac {5}{2} \left (x -2\right )-\frac {3}{8} \left (x -2\right )^{2}+\frac {1}{12} \left (x -2\right )^{3}-\frac {5}{192} \left (x -2\right )^{4}+\frac {3}{320} \left (x -2\right )^{5}-\frac {7}{1920} \left (x -2\right )^{6}+\frac {1}{672} \left (x -2\right )^{7}+\mathrm {O}\left (\left (x -2\right )^{8}\right )\right ) c_{2}+\left (c_{2} \ln \left (x -2\right )+c_{1}\right ) \left (1+\left (x -2\right )+\mathrm {O}\left (\left (x -2\right )^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 90
AsymptoticDSolveValue[x*(x-2)*y''[x]+2*(x-1)*y'[x]-2*y[x]==0,y[x],{x,2,7}]
\[ y(x)\to c_1 (x-1)+c_2 \left (\frac {1}{672} (x-2)^7-\frac {7 (x-2)^6}{1920}+\frac {3}{320} (x-2)^5-\frac {5}{192} (x-2)^4+\frac {1}{12} (x-2)^3-\frac {3}{8} (x-2)^2-2 (x-2)+\frac {2-x}{2}+(x-1) \log (x-2)\right ) \]