|UBC Calculus Online Course Notes|
Introducing a Differential Equation
Growth and Decay Phenomena
Applications of the Exponential Functions and Logarithms
By studying the exponential functions, and picking a convenient base, we have inadvertantly stumbled on a relationship satisfied by the function and its derivative. That relationship is called a differential equation . The topic of differential equations is an extremely important one in mathematics and science, as well as many other branches of studies (economics, commerce) in which changes occur and in which predictions are desirable. In most such circumstances, the systems studied come with some kind of "Natural Laws", or observations that, when translated into the language of mathematics, become differential equations. It is then the job of the mathematician to try to figure out what are the predictions, i.e. to find the functions that satisfy those equations.
In our work we have been lucky enough to start with a function and show that there is a differential equation that it satisfies, .
The function will no longer satisfy this equation. (Why not ? Try plugging it in and find out !) However, maybe if we experiment a little bit with similar functions, we will find one that works. Suppose we modify the simplest exponential and try out the possible solution
where are constants. To find whether this function satisfies the above differential equation, we will need to compute its derivative, which we do using the Chain Rule. (We need to use the Chain Rule because the function above has an expression in the exponent, not just alone.)
Notice that we have succeeded to show that the function will satisfy the equation
Thus, if we pick , i.e select as the candidate for the right function, it should indeed satisfy the differential equation in our example.
Let us summarize the important result that we have just discovered and state it in a more general form:
This is true for any value of the constant .
Why do we get so many possible solutions that all satisfy the same differential equation? Well, this stems from the fact that the differential equation is only telling us something about the way that our function is changing, and not where it "starts". If we had more information, for example, if we knew one particular point through which the "right" curve goes, we could select one of the members of the family as the single solution of interest to us. For example, if we specify the point on the y axis through which the curve passes (i.e. where is some given number) this will do the job ! Such additional information is called the initial value of the solution, since it is specifying the value of y at time .
Example Given that , find the appropriate solution of the differential equation.
Solution: Setting , and noting that we find that . Thus the "right" function, in this case, is .
The problem consisting of a differential equation together with an initial value is called an initial value problem. To summarize, we have just made a connection as follows:
For your consideration: