We can use the separation of variables technique to solve various types of growth and decay problems.

Suppose the time rate of change of y is directly proportional to the value of y, as shown by the given differential equation.

Here, k is known as the proportionality constant.

Now, suppose that at time t=0 the value of y is y0.

The value of the constant C is then y0.

This gives us a particular solution of the differential equation.

When k > 0, we have an exponential growth model.

It follows that when k < 0, we have an exponential decay model.

Suppose a population of rabbits grows exponentially.

Initially, there are 10 rabbits.

After one month, there are 15 rabbits.

The rate at which a sample of radioactive material decays can be modeled by an exponential decay model.

Lambda is known as the decay constant.

Here we have the graph of the decay of the radioactive species Iodine-131. Move the slider to vary the time to see how the material decays over time.

The amount of time it takes for radioactive material to decay to half its original amount is known as its half-life.

If we set the initial value to one, and then set the equation equal to one-half when the time is equal to the half-life, we can determine the relationship between the decay constant and the half-life.

We see that the decay constant is equal to the natural log of two over the half-life.

Let's look again at the exponential growth model for a population of rabbits. According to this model the rabbit population would continue to grow unabated.

This is not plausible, because at some point the lack of sufficient food resources would limit the population growth.

An improved model for the rabbit population growth is a bounded growth model like that given here.

Again, we start with 10 rabbits, and after one month there are 15.

We see in this graph of the resulting function that the rabbit population now approaches some upper limit.

A bounded growth model such as this one provides a more realistic prediction of the rabbit population than the exponential growth model we looked at earlier.

Now, suppose we have a bath tub full of hot water. Over time, the bath water cools.

The rate of cooling is given by Newton's law of cooling. It states that the rate at which an object's temperature changes is directly proportional to the difference between the temperature of the object, and the temperature of the surroundings.

As we shall see, this is an example of a bounded decay model.

Initially, the water in the bath tub has a temperature of 60° celsius.

The ambient temperature of the surroundings is 20° celsius.

Using Newton's law, we setup a differential equation to model the rate of change of the temperature of the bath water.

After 5 minutes the temperature of the bath water decreases to a value of 55° celsius.

Looking at a graph of the function, we see that the lower bound of the function is 20° celsius. The bath water approaches this temperature, but never quite reaches it.

What is the general solution of this differential equation? Click the "Submit" button after selecting your answer.

Which of the following models is represented by this equation when k > 0? Click the "Submit" button after selecting your answer.

How many rabbits will there be after one year? Click the "Submit" button after selecting your answer.

After how many days does the amount of Iodine-131 decay to half its original amount? Click the "Submit" button after selecting your answer.

What is the decay constant of iodine-131? Click the "Submit" button after selecting your answer.

How many rabbits will there be after one year? Click the "Submit" button after selecting your answer.

What is the general solution of this differential equation? Click the "Submit" button after selecting your answer.

After how many minutes will the temperature of the bath water decrease to a value of 40° celsius? Click the "Submit" button after entering your answer.

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