(a) The function satisfies
(i) Define the Wronskian of two linearly independent solutions and . Derive a linear first-order differential equation satisfied by .
(ii) Suppose that is known. Use the Wronskian to write down a first-order differential equation for . Hence express in terms of and .
(b) Verify that is a solution of
where and are constants, provided that these constants satisfy certain conditions which you should determine.
Use the method that you described in part (a) to find a solution which is linearly independent of .