Chapter 3.1, Problem 46E

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

$\frac{dA}{dt}=k_1(M-A)-k_{2}A$,

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

1. (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.
2. (b) Solve the DE subject to A(0) = 0. Sketch the graph of A(t) and verify your prediction in part (a).