- #1

Karagoz

- 52

- 5

Every year the population growth rate is 5% per year.

y' shows the growth rate of the y (population).

Since the population grows by 5% every year, the growth rate is:

y' = 0.05y.

This is a simple differential equation.

When y(0) = 1000

Then using a math software, the formula for the population is:

y(t) = 1000*e^(0.05t)

OR

We have a population of z = 1000 at year (1980) (call it year 0)

The population growth rate 5% per year.

Since the population grows by 5% per year, we can say:

z(t) = 1000*(1+0.05)^t = 1000*1.05^t

Derivation of z(t):

z’(t) = 1000(ln1.05)*e^(t*ln1.05)

Written as differential equation:

z’(t)=(ln1.05)*z(t)

The formula similar to z(t) is used when describing the growth of a money (in a bank at a interest rate of 5%).

Both the formula y(t) and formula z(t) describes growth rate by 5% per year.

But it’s obvious that z(t) ≠ y(t)

**What is the difference between y(t) = 1000*e^(0.05t) and z(t) = 1000*1.05^t when both describes a growth rate of 5% per year?**

**What does z(t) describe and what does y(t) describe, and what’s the difference between what each formula describe?**