Chapter 7, Problem 6RCC

a.

To determine

Write the differential equation that express the law of natural growth

Answer to Problem 6RCC

The differential equation that expresses the law of natural growth is $\frac{dy}{dt}=ky$

Explanation of Solution

Given information : The question is

To give the differential equation that express the law of natural growth

The law of natural growth is

  $\frac{dy}{dt}=ky$

The above equation means the rate of change of growth equals a constant times the population

  $\frac{1}{y}\frac{dy}{dt}$

To arrive then this quantity the equation shows in cell as follows

  $\frac{1}{y}\frac{dy}{dt}=k$

Form here it can be conclude that the quantity known as relative growth will be a constant through time

b.

To determine

At what circumstances the population growth model be an appropriate

Explanation of Solution

Given information : The question is

To explain that the population growth model be an appropriate

There are two conditions are shown

  $k>0$ , otherwise the population decreases

This model is only use when the population growth is completely natural

c.

To determine

To find : the solution of the equation

Answer to Problem 6RCC

The solution is $y=C{e}^{kt}$

Explanation of Solution

Given information : The equation is $\frac{dy}{dt}=ky$

  $\frac{dy}{dt}=ky$

Rewrite the above equation

  $\begin{array}{l}\frac{dy}{y}=kdt\\ {\displaystyle \int \frac{dy}{y}}={\displaystyle \int kdt}\\ \ln \left|y\right|=kt+C\\ y=e^{kt+C}\\ y=e^{kt}{e}^C\\ y=C{e}^{kt}\end{array}$