DiffEQ Notes

DiffEQ Notes - Day 1: In the first class, we covered what...

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Day 1: In the first class, we covered what differential equations are, what makes them linear or not linear, and why we care. Differential equations have equal signs. Differential equations also have "fleas." (The prime, indicating a derivative.) The first question that one asks on seeing a differential equation is what the independent and dependent variables are. You can tell which is the dependent variable is by looking for the "flea" (the derivative sign). A differential equation is an ordinary differential equation (the only kind that we will study in Math 308) if it has only one independent variable. (There are also courses in partial differential equations. They deal with situations where there are two independent variables, eg., the temperature reading on a metal rod depends both on where you measure and when you measure, ie., on two independent variables.) The reason that one wants to know which is the dependent variable is that what is done with the dependent variable in the equation determines if the equation is linear or nonlinear. In general, if the equation is nonlinear, with very few exceptions, no one knows how to give a formula solution. The best that one can hope for is a qualitative description of solutions, with perhaps numeric approximations generated on a computer. If the given differential equation is linear, there is hope that a formula solution may be found. You should decide up front which kind of equation you have and what approach you are going to take. NSS shows in p. 5 (7) the form that a differential equation must have in order for the equation to be linear. In words, the only things that one can do with the dependent variable and its derivatives is to multiply them by arbitrary functions of the independent variable and add or subtract them. In particular, if the dependent variable is y, then the equation is nonlinear if it contains expressions like yy', y^2, 1/y, sqrt(y''), log(y), sin(y), exp(y'), etc., which are not simply multiplying y, y', y'', etc. by a known function of x and adding up the terms. Day 2: At the beginning of class, we practiced opening a Maple worksheet, writing mathematics, and putting our name, UIN, and section number in a paragraph that we inserted before the mathematics. When you work on a quiz or exam, you can always insert work in the
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middle of other work, so that the problems come out in numerical order. That helps me to grade the papers quickly and get them back to you. We used both Save As and Export As Mapletext to save it. During an exam, you will be asked to Export As Mapletext so that if the network crashes, most of the work will still be saved; and you will only lose a few minutes time, rather than having to start over. We also practiced printing to a postcript file, out.ps, and using the command turnIn out.ps
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DiffEQ Notes - Day 1: In the first class, we covered what...

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