Question

Let the rate of growth $\frac{\mathbf{d}\mathbf{N}}{\mathbf{d}\mathbf{t}}$ of a colony of bacteria be proportional to the square root of the number present at any time. If there are no bacteria present at t=0 , how many are there at a later time? Observe here that the routine separation of variables solution gives an unreasonable answer, and the correct answer, N=0 , is not obtainable from the routine solution. (You have to think, not just follow rules!)

Short answer

The solution is$N\left(t\right)=0$

Step-by-step solution

Given Information.

It is given that the rate of growth $\frac{dN}{dt}$ of a colony of bacteria be proportional to the square root of the number present at any time.

Definition of Differential equation

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

Solve differential equation

The differential equation given is

$$\begin{gathered} \frac{dN}{dt}=\sqrt{N\left(t\right)} \\ \int \frac{dx}{\sqrt{N\left(t\right)}}=dt \end{gathered}$$

Therefore, the general solution is

$N\left(t\right)={\left(\frac{t-t_0}{2}\right)}^2$

Where$t_0$is the integration constant

It is given that at t=0 , N(t)=0 .

For this condition, no solution of differential equation is there for any value of t=0

Since at t=0 , given N(t)=0 , so after any time the number of the bacteria will remain zero, so the solution will be the singular solution which is N(t)=0 . This solution can not be found out by separation of variables because $\sqrt{N\left(t\right)}$ is in the denominator.

Therefore, the solution is N(t)=0.