Internal problem ID [6248]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page
391
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 )^{2} y^{\prime \prime }-2 \left (x -2\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: 0.063 (sec). Leaf size: 55
Order:=8; dsolve(x*(x-2)^2*diff(y(x),x$2)-2*(x-2)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=2);
\[ y \relax (x ) = \left (x -2\right ) \left (\left (-\frac {1}{2} \left (x -2\right )+\frac {1}{8} \left (x -2\right )^{2}-\frac {1}{24} \left (x -2\right )^{3}+\frac {1}{64} \left (x -2\right )^{4}-\frac {1}{160} \left (x -2\right )^{5}+\frac {1}{384} \left (x -2\right )^{6}-\frac {1}{896} \left (x -2\right )^{7}+\mathrm {O}\left (\left (x -2\right )^{8}\right )\right ) c_{2}+\left (c_{2} \ln \left (x -2\right )+c_{1}\right ) \left (1+\mathrm {O}\left (\left (x -2\right )^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.029 (sec). Leaf size: 90
AsymptoticDSolveValue[x*(x-2)^2*y''[x]-2*(x-2)*y'[x]+2*y[x]==0,y[x],{x,2,7}]
\[ y(x)\to c_1 (x-2)+c_2 \left (\left (-\frac {1}{896} (x-2)^7+\frac {1}{384} (x-2)^6-\frac {1}{160} (x-2)^5+\frac {1}{64} (x-2)^4-\frac {1}{24} (x-2)^3+\frac {1}{8} (x-2)^2+\frac {2-x}{2}\right ) (x-2)+(x-2) \log (x-2)\right ) \]