205sp11-11-14 - MATH 205 SPRING 2011 Math 205 is really a...

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MATH 205, SPRING, 2011 Math 205 is really a combination of two courses, linear algebra and di erential equations. The main reason for this is so that you don't have to take yet another math course, but there are some sound reasons for it as well. Di erential Equations is the basic tool for mathematical modeling of physical systems. The idea is to turn basic understanding of the forces at work in a system into a precise description of the resulting behavior. This is the reason why engineers need to study calculus. On the other hand, Linear Algebra can be thought of as merely a set of formal manipulations and theory. However, it is a central part of the modeling process we need to do di erential equations. It's practically impossible to understand how you work with di erential equations without using linear algebra. In the nal analysis, though, both of these subjects will be very useful for your later course work. 1. First-Order Equations 1.1. Introduction. I should start o with an explanation of what a di erential equation is. A di erential equation is an equation (gee!) involving some unknown function, usually called y , maybe some known functions of the variable (usually x ), and derivatives of the function y . A simple example might be: y 00 + 5 y 0 - 17 y = e x . Many texts tend to write a general equation in a confusing way: F ( x, y, y 0 , . . . , y ( n ) ) = 0 . They just mean to say that it is an equation (the = ) involving x , y , y 0 , and so on, up to the n th derivative y ( n ) . The F refers to the expression involving all these terms it's not the function that solves the equation. Our text sometimes uses what may seem to be a di erent general expression, d n y dx n = f ( x, y, y 0 , ..., y ( n - 1) ) , but really that's the same sort of equation, but solved for y ( n ) . Since if y ( n ) occurs nontrivially (not multiplied by 0, which would be pretty trivial), it can be solved for, at least in theory. The point is to nd a solution of the equation, which is a function y = y ( x ) which ts into the equation. For example, you can check that the function: y = - e x / 11 does solve the equation in the previous paragraph. However, that isn't the only function that solves that di erential equation. The functions: y = 2 e (( 93 - 5) x/ 2) - e x / 11 and y = 3 e (( - 93 - 5) x/ 2) - e x / 11 also solve the equation. Don't despair at those stupid 93 's. You'll see, soon, how those got there. And. you'll see that it wasn't hard to nd those solutions, how they are all related, and why I knew how many di erent solutions there might be. For now, you should be able at least to check that the functions do indeed solve the equation. There is a dirty little secret here. I didn't really solve that equation by hand. I used a computer-algebra
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This note was uploaded on 04/16/2011 for the course MATH 205 taught by Professor Zhang during the Spring '08 term at Lehigh University .

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205sp11-11-14 - MATH 205 SPRING 2011 Math 205 is really a...

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