i need some help....?

Question

can anyone figure this out?
In the absence of environmental constraints, population grows at a rate proportional to its size. Express the rate of population growth as a function of the size of the population.

Answer 1

If P(t) is the population at time t, the rate of growth is dP/dt, which is proportional to P, so dP/dt = k P, where k is the proportionality constant.

Answer 2

dp/dt=kp, p=population
dp/p=kdt, we have separated the variables;
integrating;
lnp=kt+C, C=constant
let p=p(0), whent=t(0)
ln(p(0))=C
so ln(p/p(0))=kt
p=p(0)e^(kt). AN.S.