Section 1-3 : Final Thoughts
Before moving on to learning how to solve differential equations we want to give a few final thoughts. Any differential equations course will concern itself with answering one or more of the following questions.
- Given a differential equation will a solution exist?
Not all differential equations will have solutions so it’s useful to know ahead of time if there is a solution or not. If there isn’t a solution why waste our time trying to find something that doesn’t exist?
This question is usually called the existence question in a differential equations course.
- If a differential equation does have a solution how many solutions are there?
As we will see eventually, it is possible for a differential equation to have more than one solution. We would like to know how many solutions there will be for a given differential equation.
There is a sub question here as well. What condition(s) on a differential equation are required to obtain a single unique solution to the differential equation?
Both this question and the sub question are more important than you might realize. Suppose that we derive a differential equation that will give the temperature distribution in a bar of iron at any time \(t\). If we solve the differential equation and end up with two (or more) completely separate solutions we will have problems. Consider the following situation to see this.
If we subject 10 identical iron bars to identical conditions they should all exhibit the same temperature distribution. So only one of our solutions will be accurate, but we will have no way of knowing which one is the correct solution.
It would be nice if, during the derivation of our differential equation, we could make sure that our assumptions would give us a differential equation that upon solving will yield a single unique solution.
This question is usually called the uniqueness question in a differential equations course.
- If a differential equation does have a solution can we find it?
This may seem like an odd question to ask and yet the answer is not always yes. Just because we know that a solution to a differential equations exists does not mean that we will be able to find it.
In a first course in differential equations (such as this one) the third question is the question that we will concentrate on. We will answer the first two questions for special, and fairly simple, cases, but most of our efforts will be concentrated on answering the third question for as wide a variety of differential equations as possible.