A certain fish population in a secluded lake is governed by the differential equation…

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A certain fish population

P(t)

in a secluded lake is governed by the differential equation

\frac{dP}{dt}=\frac{P(P-200)}{t},\qquad t>0,

where

P

is measured in hundreds of fish and

t

in weeks. (1) Part 1: Derive a relation between

P

and

t

that represents the general solution to the differential equation. (2) Part 2: Given the initial condition

P(1)=100

, obtain an explicit expression for

P

as a function of

t

valid on the maximal interval containing

t=1

for which the solution is defined. (3) Part 3: Classify the long-term behavior of the population by evaluating

\displaystyle\lim_{t\to\infty}P(t)

and interpreting your result in the context of the lake ecosystem.