Internal problem ID [6162]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page
355
Problem number: 27.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (x -2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 2\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 34
Order:=8; dsolve(diff(y(x),x$2)+(x-2)*y(x)=0,y(x),type='series',x=2);
\[ y \relax (x ) = \left (1-\frac {\left (x -2\right )^{3}}{6}+\frac {\left (x -2\right )^{6}}{180}\right ) y \relax (2)+\left (x -2-\frac {\left (x -2\right )^{4}}{12}+\frac {\left (x -2\right )^{7}}{504}\right ) D\relax (y )\relax (2)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 51
AsymptoticDSolveValue[y''[x]+(x-2)*y[x]==0,y[x],{x,2,7}]
\[ y(x)\to c_1 \left (\frac {1}{180} (x-2)^6-\frac {1}{6} (x-2)^3+1\right )+c_2 \left (\frac {1}{504} (x-2)^7-\frac {1}{12} (x-2)^4+x-2\right ) \]