Question

In one model of the changing population P(t) of a community, it is assumed that dP dt = dB dt − dD dt , where dB/dt and dD/dt are the birth and death rates, respectively. (a) Solve for P(t) if dB/dt = k1P and dD/dt = k2P. (Assume P(0) = P0.)

Answer 1

Answer:

$P(t) = P_{0}exp^{(k_{1}-k_{2})t$

Step-by-step explanation:

$\frac{dP}{dt}= \frac{dB}{dt}-\frac{dD}{dt}\\where \frac{dB}{dt}=k_{1}P ; and \frac{dD}{dt}=k_{2}P\\ Therefore:\\\frac{dP}{dt}=k_{1}P-k_{2}P\\\frac{dP}{dt}=(k_{1}-k_{2})P$

This leads to:

$\frac{dP}{P}=(k_{1}-k_{2})dt$

Taking the integral of both sides

$\int {\frac{dP}{P}=\int(k_{1}-k_{2})dt} \\ln P = (k_{1}-k_{2})t+C, C=constant of integration$......(i)

$P(t) = Cexp^{(k_{1}-k_{2})t$

When t=0, P(t)=$P_{0}$

From (i), $P_{0} = C$

Therefore: $P(t) = P_{0}exp^{(k_{1}-k_{2})t$