The number of individuals in a population of bacteria is described by the follow

Question

The number of individuals in a population of bacteria is described by the following differential equation: dP(t) =kP(t) (i) where P(t) is the population, t is the time in hours, and k is a constant that depends on the type of bacteria. (a) Show that the general solution of (1) is: P(t) = Ct' CD where C is mi arbitrary constant. (b) Assume that a particular population of bacteria has 500 individuals at time equal zero, and 1500 individuals after three hours. Find the particular solution that mathematically describes this situation (i.e., find the values of C and k in (2) And write the resulting equation).

Explanation / Answer

Solution : given dP / dt = k P using variable seperable method

dP / P= k dt integrating on either side

log P = kt + logC => P = C e kt is the general solution ------(1)

given t =0 P = 500 and t =3 , P = 1500 substituting in equation (1) we get

500 = C and 1500 = C e3k put C =500 => 1500 = 500 e3k  => e3k = 1500/500 =3 => 3k = log3

and k = (log3) / 3 hence the particular solution is P = 500 e( t log3) /3

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