6.2.3 Symbolic Solutions 2: Calculating
         and Checking

It may seem that we did a lot of work on the preceding page to find out something we already knew. But think about what we accomplished: We solved `(dP)/(dt)=kP` without having to guess the answer — or even the form of the answer. Thus we can hope to use this technique, called separation of variables, to solve differential equations and initial value problems in situations where there is no hope of guessing the answer — for example, the constrained growth model. We will return to the solution of that problem in the next section. Here we concentrate on further developing our technique.

As long as we were operating in the "guess the answer and check" mode, there was little doubt about correctness of the answer, because checking it was part of the process of finding it. Now that we are about to move beyond answers we can guess, we also have to pay attention to establishing the correctness of our answers. In most cases where "answer" means the result of antidifferentiation, this is not difficult, because checking can be done by the easier of the two inverse processes, differentiation. However, you need to be aware that the resulting equation will probably have to be manipulated algebraically to make it look like the original differential equation.

Pretending for a moment that we do not already know that `P=P_0e^(kt)` is a correct solution to `(dP)/(dt)=kP`, we illustrate the checking process with a short calculation:

Checkpoint 1

In the following Activity, we ask you to apply the separation of variables technique to another differential-equation-with-initial-value problem and then check your answer.

Activity 4

1. Use the separation of variables technique to solve the initial value problem:

   ${\displaystyle \frac{dP}{dt}=k{P}^2,P\left(0\right)=P_0\mathrm{.}}$

2. Check your answer to part (a) by showing that your solution function satisfies both parts of the initial value problem.

Comment on Activity 4

Checkpoint 2