Let represent the size of a population governed by the differential equation dy dt =k…

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

Let

y(t)

represent the size of a population governed by the differential equation

\frac{dy}{dt}=k\,y\,(M-y),\qquad k>0,\; M>0,

with initial value

y(0)=y_0

. (a) Show that

y=0

and

y=M

are equilibrium solutions. (b) Assume

0<y_0<M

. Prove, without finding an explicit formula for

y(t)

, that

\displaystyle\lim_{t\to\infty}y(t)=M

. (c) Assume

y_0>M

. Prove that

y(t)

approaches

M

from above as

t\to\infty

. (d) Demonstrate that the population grows at its maximum instantaneous rate when

y=\dfrac{M}{2}