# differential equations: bacteria growing in a lab culture

#### turkey

##### New member
Bacteria in a lab culture grow in a way that the rate of change of bacteria is directly proportional to the number of bacteria present.

1. Write a differential equation that represents the relationship.

2. Solve the equation for the number of bacteria as a function of time.

3. Suppose that there were initiallly 4 million bacteria. Three hours later there are 6 million. Find the particular equation that expresses the number of millions of bacteria as a function of time.

#### skeeter

##### Elite Member
you should have already seen this in calculus class ... and you should be familiar with the solution equation from your precalculus course.

rate of change of bacteria is directly proportional to the number of bacteria present.
the differential equation for natural exponential change is ...

$$\displaystyle \frac{dy}{dt} = ky$$

the solution of this differential equation is ...

$$\displaystyle y = y_0 e^{kt}$$, where

y = population at any time t
y<sub>0</sub> = initial population at time t = 0
k = growth (or decay) constant.

I'll let you complete part 3.