The general solution to dy/dt = ky is y = Dekt. Here we prove that every solution to the differential equation, defined on an interval, is given by the general solution.

(a) Show that if y(t) satisfies the differential equation, then d/dt (ye^−kt) = 0.
(b) Assume that y(t) satisfies the differential equation on an interval I. Use the result from (a) and the Corollary to the Mean Value Theorem in Section 4.3 to prove that on I, y(t) = De^kt for some D.