In the Mathematical Background on CD, we prove a general existence and uniqueness result for continuous dynamical systems. The proof includes showing that Euler's approximation converges as the step size tends to zero. Mathematical uniqueness of the solution to an initial value problem is what makes dynamical systems deterministic scientific models. Uniqueness is really what you are thinking of when you say, "The solution of this system models ...." Uniqueness is also mathematically important. For now, we will use the following result:

Theorem CD-8.1 Unique Solution to a Linear Dynamical System For any real constants Y₀ and k, there is a unique real function y[t] defined for all real t satisfying

This function can be written y[t]=Y₀e^kt.

Simply put, the theorem means that saying y[t]=Y₀e^kt is exactly the same thing as saying y[0]=Y₀ and . The theorem assures us in particular that a function of our official definition exists. We know intuitively from our experience with programs like SecondSIR and EulerApprox that the computer approximations do converge. The next subsection shows how the important addition formula for exponentials follows from uniqueness of the solution to our official definition.

Let , and consider the function . We know y[0]=C and by the Superposition Rule for differentiation that . This means that y is a solution to

Now consider the function . Again, and the Chain Rule says , so z is also a solution to

This is the same dynamical system, written with a different letter. Uniqueness means that both y[t] and z[t] are the same function

Our new definition causes us a technical problem. We know how to compute rational powers of e. We want to show that the new definition extends what we already know.

Problem CD-8.1

Let e be the number Exp[1], that is, the value of the solution of the dynamical system at time 1.

• 1. Show that e²=Exp[2].
• 2. Show that e^1/3=Exp[1/3] by letting a=Exp[1/3] and computing .
• 3. Show that e^q=Exp[q] for any rational q=m/n.