#5 all parts a b c d e f g

5. Consider the constant coefficient delay-differential equation = ax(t to). (a) Plug x = ert into the ODE and get an algebraic equation relating r, a, to, which when satisfied guarantees that er is a solution. In the parts below, we will use this to characterize the solutions to the ODE. (b) Show that if a > 0, there is always a unique r, with 0 -a, then there are no real solutions of the form e". (Hint: if you have two curves f(r),g(r), then they are tangent at ro if f(ro) = g(ro) and f'(ro) = g'(ro).) (e) Now assume that r = a + Bi, and we will look for complex solutions of the form ert. Derive the equations that a and B must solve, using your equation from part a). (f) It is not obvious, but the above equations do have a solution for a,. Assume there is a value to such that a 0 for t > te Show that te = -2 (g) It is not hard to show that if a + Bi is a solution to your equations from part e), then so is a - Bi. This means that eat cos(Bt) and eat sin(Bt) are solutions to the ODE. Graph these solutions for -1 - 2