Analytic Methods

In this section, you will be solving three differential equations by the method of separation of variables. The first one is done for you so that you can see the steps to follow.

We will begin with the simplest of the equations, the one leading to the exponential model of growth or dacay: . First, we separate the variables,

,

then we integrate both sides and exponentiate to get the solution:

or

The constant is determined by an initial condition . That is, . We know the resulting equation for very well. In the case of a growing population, and the model represents a population that is growing exponentially without bound.

You should proceed now to use the method of separation of variables to solve the second equation .

Finally, you should solve the Logistic equation . To do this you will need to know two things that we have not studied yet. The first is a decomposition of a product of fractions into a sum (check it by combining the fractions on the right-hand-side). And the second is a differentiation formula that is an instance of the chain rule.

When you have solved the equation, you should also explain what happens as , assuming