When forgetfulness is taken into account, the rate of memorization of a subject is given by

dA/dt = k₁(M - A) - k₂A,

where k₁ > 0, k₂ > 0, A(t) is the amount memorized intime t, M is the total amount to be memorized, and M - A is the amount remaining to be memorized.

(a) Since the DE is autonomous, use the phase portrait concept of Section 2.1 to fi­nd the limiting value of A(t) as t → ∞. Interpret the result.

(b) Solve the DE subject to A(0) = 0. Sketch the graph of A(t) and verify your prediction in part (a).