1. A differential equation is an equation involving an unknown function and one or more of its derivatives. The function f(r) is said to be a solution to the differential equation if the equation is satisfied when y = f(r) and its derivatives are substituted into the equation. - (a) Show that the function y(t) = y(0)ekt satisfies the differential equa- dy tion dt = ky. (b) Suppose a population grows at a rate proportional to its current size; that is, the population P is described by the differential equation dP dt kP. If the population grows at a relative rate of 2% per year (so that k=0.02) and the initial size of the population is P(0) = 1000, what is the size of the population after 5 years? 2. For each of the following functions f(r), compute the derivative f'(x). (a) f(r) arctan(tan (r)) (b) f(x) = arcsin (sin(z) - cos(x)) (c) f(r)=arcsin(sin(r)) 3. For constants c and 2, show that y = c sinh(r)+c cosh(z) is a solution d'y to the differential equation