When we say the population changes at a rate proportional to its size.....?

P(t)=Poe^(kt) Is the population at time t, dP/dt, the rate in change in population, is proportional to its initial population, and anytime there after. It decreases/increases relative to the population. I suppose a proof would be adding the function P to its derivative and setting it equal to zero...a homogenous equation versus a heterogeneous equation will vary solely on P, as opposed to some input function for 0