Consider the linear homogeneous differential equation 2 14y Ay a2 dt (i) Assume that the power series y(t) = 2::]:0 c,t" with a radius of convergence p 0 is a solution to the above differential equation. Show that the coefficients , satisfy g R and for n I, the recursio formula Cptl = n+1 holds. Now, let y; () correspond to the solution of the differential equation where y;(0) = 1 and %{(0) = 1. (i) Provide a series representation for the solution y; () and conclude the function that the series corresponds to. (ili) Construct, using y; (), a solution y,(t) such that the complete homogeneous solution to the differential equation can be written as the linear combination y(t) = cru (t) + oy (t), for arbitrary constants ; and