Consider a population of bacteria that grows according to the initial value problem dP/dt=P/10, P (0) = 300. Find the population size after 40 hours

dP / dt = P / 10

We apply separable variables:

dP / P = dt / 10

We integrate both sides:

Ln (P) = t / 10 + C

We rewrite the equation:

Exp (Ln (P)) = Exp (t / 10 + C)

P = Exp (C) * Exp (t / 10)

P = C * Exp (t / 10)

We look for the constant using:

P (0) = 300

300 = C * Exp (0/10)

300 = C * 1

C = 300

We rewrite the equation:

P = 300 * Exp (t / 10)

After 40 hours we have:

P = 300 * Exp (40/10)

P = 16379.44501

Answer:

the population size after 40 hours is:

P = 16379