Question

Suppose the rate at which bacteria in a culture grow is proportional to the number present at any time. Write and solve the differential equation for the number N of bacteria as a function of time t if there are ${\mathit{N}}_{\mathbf{0}}$bacteria when $\mathit{t}\mathbf{=}\mathbf{0}$. Again note that (except for a change of sign) this is the same differential equation and solution as in the preceding problems.

Short answer

Answer

The number N of bacteria as a function of time t is $N\left(t\right)=N_0{e}^{kt}$

Step-by-step solution

Given information

It is given that the rate at which bacteria in culture grow is proportional to the number present at any time, and there are $N_0$bacteria when$t=0$.

Definition of differential equation

A differential equation is an equation that contains at least onederivative of an unknown function, either an ordinary derivative or a partial derivative.

Find N as a function of t

It is given that the rate at which bacteria in culture grow is proportional to the number present at any time.

Therefore,

$$\begin{gathered} \frac{dN}{dt}FkN \\ \int \frac{dN}{N}FKd\int t \\ \alpha \left(N\right)=t+ \\ \alpha N\left(et\right){=}_1{}^{kt} \end{gathered}$$.

Now at $t=0,N=N_0$

$$\begin{gathered} \alpha N\left(e0\right)=N_0={}_1{}^0 \\ \Rightarrow {\alpha }_1=N_0 \end{gathered}$$

Thus,

$N\left(t\right)=N_0{e}^{kt}$.

Therefore, the number N of bacteria as a function of time t is $N\left(t\right)=N_0{e}^{kt}$.