To determine

(a)

Write a differential equation that express the law of natural growth.

Solution: If dydx=ky, then y=cekt.

Given: The law of natural growth.

Explanation

If  dydx=ky, then y=cekt.

Where c is the initial amount present.

K is either a growth or decay constant.

K is positive: equation represents growth.

K is negative: equation represents decay.

Conclusion: Hence, if dydx=ky then y=cekt for law of natural growth.

(b)

To determine: Under what circumstances is this an appropriate model of population growth.

Solution: Exponential growth, biotic potential, environmental resistance.

Given:  Appropriate model of population growth.

The circumstances is given below:

Exponential growth: - Number of individuals added each generation increases as the total number of family increases.

Biotic potential: - Biotic potential is maximum population growth that can possibly occurs under ideal circumstances.

Environmental resistance: - Environmental resistance is all environmental condition that prevent population from achieving biotic potential.

Conclusion: Hence, exponential growth, biotic potential and environmental resistance are under circumstance.

(c)

To determine: What are the solution of the equation, dydx=ky, y=cekt.

Solution: kcekt.

Given: dydx=ky and y=cekt.

y=cekt

ddt(cekt)=kcekt=kcekt

Conclusion: Hence, the solution is kcekt.