#### 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.