Section Summary

In this section we considered the constrained growth model for population growth and realized that our previous guess-and-check method of attacking initial value problems would not work. Instead, we introduced the separation-of-variables approach, which requires rewriting the differential equation as an equality of differentials with all occurrences of the dependent variable on one side and all occurrences of the independent variable on the other. We look for antidifferentials of the two sides, and when we find them, we know that they must differ at most by an additive constant. We use the initial condition to determine the value of the constant.

At this point we have reduced our problem to an algebraic computation to solve for the dependent variable as an explicit function of the independent variable. We note in passing that the algebraic step is not always possible -- sometimes we have to leave the result as an implicit definition of the solution function. However, for the time being, we will consider only problems for which explicit solutions are possible.

Since checking is not part of this method, we need to confirm our result by differentiating the solution function to show that it satisfies the differential equation, and we also need to check that it satisfies the initial condition.

In the next section we will apply this technique to the constrained growth model.