2.7: All of the words about existence and uniqueness.

Section 2.7 is entitled “Existence and Uniqueness of Solutions.”

When dealing with initial value problems, we could ask ourselves two questions:

1. When can we be sure a solution exists at all?

2. How many different solutions are there?

If we were to have these questions, we would be questioning existence and uniqueness (see what I did there?).

Let’s answer these questions in order by starting with the issue of existence.

Sometimes initial value problems don’t make sense at all. You’ll be solving your problem without a care in the world and the problem will take a left turn into Worry And Devastation and will ask you to divide by zero or put a negative number into a natural log. In any event, there would be no solution to your initial value problem and that’s sad.

The book calls these types of initial value problems “anomalies,” especially when they are in normal form (reminder: x’ = f(t, x) is normal form). 

Nevertheless, here’s a theorem on the existence of an initial value problem:

Suppose a function f(t, x) is defined and continuous on a rectangle R in the tx-plane. This means at any point (t₀, x₀) that is within that rectangle, the initial value problem x^’ = f(t, x) and x(t₀) = x₀ has a solution defined that contains that t₀. This solution will be defined until the curve leaves that rectangle. Note that it is required for the equation to be written in normal form for this theorem to stand. 

Back in section 2.1, we defined the interval of existence as “the largest interval in which the solution can be defined” (78). Note that the interval of existence cannot usually be determined from the existence theorem. Instead, you will have to find the interval by finding an explicit solution. 

Let’s focus specifically on linear equations. Recall we defined linear equations to have the form x^’ = a(t)x + g(t). This would make our normal form f(t, x) for linear equations a(t)x + g(t). If a(t) and g(t) are continuous for a certain interval of t, this means f is continuous on a rectangle that as that same interval of t and -∞ < x < ∞. This means the existence theorem can guarantee the general solution exists over that entire interval of t. Yay linear equations!

Our existence theorem said that f(t, x), our right hand side of the equation, must be continuous. Sometimes that isn’t the case and we have a right hand side that isn’t continuous. However, we still want to solve an initial value problem. Although these cases do not satisfy our theorem, we want to break the rules and be rebels and get ourselves an answer. In most of these cases, the equation is linear [meaning x^’ = a(t)x + f(t)] and the only discontinuity is in f(t). We can agree that the solution x(t) that we find will be continuous and satisfy everywhere except where f(t) is discontinuous.

Moving onto our second question, which asked how many solutions are there. In this case, we’re talking about uniqueness. Let us start with new vocabulary:

If there’s only one solution to a system, then the system acts the exact same way if it’s started from the same initial conditions. This system is then called deterministic. If there are multiple solutions, then the system is unpredictable. Thus for an initial value problem to have a unique solution, it must also be deterministic.

In the book, there is quite a bit of exposition leading into the theorem. I’m not including the exposition because it’s basically proving the theorem and it includes hypotheses and conclusions and stuff. Because this is technically a summary, we don’t need silly hypotheses and exposition. 

Anyway, onto the theorem:

“Suppose f(t, x) and its partial derivative δf/δx are both continuous on the rectangle R in the tx-plane. Suppose (t₀, x₀) [is located on that rectangle R] and that the solutions x^’ = f(t, x) and y^’ = f(t, y) satisfy x(t₀) = y(t₀) = x₀. Then as long as (t, x(t)) and (t, y(t)) stay in R, we have x(t) = y(t)” (82). 

The short version of all that says two solutions to the same equation that start in the same place will travel together. In other words, any particular point that is contained on R will only have one solution curve. Note that any point on R can be the starting point.

Obviously, we will want to apply these theorems to initial value problems. It will be necessary, however, to find a rectangle R in which the equation satisfies the hypothesis.

The uniqueness theorem also has a geometric interpretation. We’ll be thinking in terms of the solution curves for this interpretation. If we have two functions that satisfy the same point, then those functions meet at that point. Additionally, if these functions are solutions to the same differential equation, then the uniqueness theorem says that the functions are equal for all points. This would mean that the graphs would coincide, but solution curves cannot meet. This means that they cannot cross or touch each other.

Solution curves can get exponentially close to one another, but the uniqueness theorem states they never meet. Don’t let computer drawn pictures fool you, then. If you zoom in far enough, you will see that these curves do not meet. 

Extra sources: http://www.math.uiuc.edu/~tyson/existence.pdf