(a) Define the Wronskian of two solutions and of the differential equation
and state a necessary and sufficient condition for and to be linearly independent. Show that satisfies the differential equation
(b) By evaluating the Wronskian, or otherwise, find functions and such that has solutions and . What is the value of Is there a unique solution to the differential equation for with initial conditions ? Why or why not?
(c) Write down a third-order differential equation with constant coefficients, such that and are both solutions. Is the solution to this equation for with initial conditions unique? Why or why not?