MAT 242 Lecture 1

1.1 Wherein we introduce some of the basic ideas in the study of Differential Equations.

» A differential equation is one which contains a derivative. You have probably seen one or two elementary ones in your journey through Calculus-land, like, say:

,

Note the presence of the derivative , and the affixed initial condition .

» When solving a DE, the object is to find the function satisfying the equation. In the above example, we would like to find the function f whose derivative is 2x and which also meets the specified initial condition. In this case, we reach our objective by simply integrating both sides of the equation, then solving for the integration constant C using the IC (work in class).

Pretty straightforward. But things ain't always so easy - DE's can appear in much more complex form. And to each form corresponds a different solution technique. Some techniques may combine other techniques within them. Some techniques may be technically intricate and all demand accuracy to bring home the proper solution. So try and keep your cool and stay focused. Practice won't hurt, and remember that concepts point to strategies. I hope this last sentence will make sense to you at some point during the course (for your sake and mine).

» And so, in quick, cold, and corporate style, I would describe this course as "the presentation of classes of differential equations, and techniques for their solution. This will necessarily include a discussion of theory of DE's (and their solutions)."

Of course, the devil's in the details.

Ex: Show that y = cos(x) is a solution to the differential equation . Discuss the differences between this DE and the 1st one discussed.

» Keepin' it real. Mathematicians like to at least pretend that what they're doing is connected to reality (that's a joke on different levels). Thus math is used to model the physical processes present 'out there'. It is important here to revisit a crucial concept developed in Calculus, which, presented in a sort of chant-form would be "derivative is slope is change". That's right - when we talk about change of some process, we 'hard-code' that in the language of mathematics as a derivative. There are many 'keywords' that connote 'change' within the study of a process. Your text gives some examples in a passage on p. 1: "the motion of fluids, the flow of current in electric circuits, the dissipation of heat in solid objects, the propagation and detection of seismic waves, or the increase or decrease of populations...". (Discuss, circle the words that mean 'change') In addition, known physical laws may be incorporated into models. Let's take a look at an example:

do: Ex 1, p. 2.

» Slope Fields. There are some special graphs we use to study solutions of differential equations. The first of these is the slope field, which is constructed in the plane with values of t (or the independent variable) on the x-axis, y (or whatever function we're trying to solve the DE for) on the y-axis and at points (t, y) , we draw little line segments with slope equal to the value of when we plug (t, y) into the equation.

Ex: Ex. 2, p. 3

Note, from this example, that we can draw qualitative conclusions about the solutions, without even solving the DE. We can see that the velocity ultimately tends ('points') toward the equilibrium solution.

» As the text notes, we often use computers or graphic calculators to draw slope fields. I will be making use of one called MYSLOPE on the TI-83, and will transfer it to your calculators as well. Let's try an example with it, example 3 on p. 6. We want to draw a slope field for the predator-prey equation

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We start by entering the equation into Y2 , in . Here, Y2 is playing the role of the derivative, in this case, . Also, we use the variable Y instead of p. Now, you do have to do some work. In order to get a relevant view of the direction field, we need to know what the equilibrium solution is, and center the settings on it (on the y-axis). So set and solve:

So the y-values should be centered around y = 900. I'm going to follow the text's lead:

Now, hit and select MYSLOPE, hitting :

, , , and we have our slope field.

Note what the slope field is indicating: if the mouse population starts below the equilibrium, they all get eaten. If it rises above the equilibrium, the population keeps growing indefinitely.