(a) By using a power series of the form
or otherwise, find the general solution of the differential equation
(b) Define the Wronskian for a second order linear differential equation
and show that . Given a non-trivial solution of show that can be used to find a second solution of and give an expression for in the form of an integral.
(c) Consider the equation (2) with
where and have Taylor expansions
with a positive integer. Find the roots of the indicial equation for (2) with these assumptions. If is a solution, use the method of part (b) to find the first two terms in a power series expansion of a linearly independent solution , expressing the coefficients in terms of and .