Consider the differential equation
What values of are ordinary points of the differential equation? What values of are singular points of the differential equation, and are they regular singular points or irregular singular points? Give clear definitions of these terms to support your answers.
For not equal to an integer there are two linearly independent power series solutions about . Give the forms of the two power series and the recurrence relations that specify the relation between successive coefficients. Give explicitly the first three terms in each power series.
For equal to an integer explain carefully why the forms you have specified do not give two linearly independent power series solutions. Show that for such values of there is (up to multiplication by a constant) one power series solution, and give the recurrence relation between coefficients. Give explicitly the first three terms.
If is a solution of the above second-order differential equation then
where is an arbitrarily chosen constant, is a second solution that is linearly independent of . For the case , taking to be a power series, explain why the second solution is not a power series.
[You may assume that any power series you use are convergent.]