diff_eq - Electricity and Magnetism I (P331) M. R. Shepherd...

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Unformatted text preview: Electricity and Magnetism I (P331) M. R. Shepherd October 14, 2008 Differential Equations The purpose of this note is to provide some supplementary background on differential equations. The problems discussed below deal largely with those we have encountered in introductory mechanics. Hopefully a discussion of differential equations in the context of mechanical systems, where we are likely to have a more solid physical intuition, will let us focus on the math itself. Once we have a conceptual understanding of the mathematics then it may be useful to review the problems in the context of electrostatics as discussed in the text. There have been many textbooks written on both ordinary and partial differential equations most math students take a full year to study each topic. Therefore, it is impossible to teach the topic in just a few pages, but hopefully the notes below provide a brief introduction to a couple of elementary differential equations frequently encountered in physics and give some insight on the general technique for solving them. Ordinary Differential Equations A differential equation is an equation that describes how a function changes . In the case of a typical equation that most of us learned about in our first algebra class, we search for a single value that is the solution. For example the solution to x + 1 = 10 is x = 9. With differential equations, we are instead looking for a function that satisfies the equation instead of just a numerical value. When that function is a function of just one variable we say that the differential equation is an ordinary differential equation (ODE). Motion with constant acceleration Lets take a familiar example from mechanics. Suppose we have a object with constant acceleration a . We can write dv dt = a. (1) Here v is really v ( t ), a function of time. In this example a is a constant. The general solution to this equation is v ( t ) = at + C, (2) where C is an arbitrary constant. It turns out in the context of our physical problem at time t = 0 we have v (0) = C and therefore we associate C with the initial velocity v and write v ( t ) = v + at. (3) This equation above is also a differential equation that describes the rate of change of x ( t ). We have dx dt = v + at, (4) to which the general solution is x ( t ) = v t + 1 2 at 2 + C. (5) Similar to above, when t = 0 we have x (0) = C and we associate C with x , the initial position of the object. These two equations may hardly seem like differential equations. They are quite easy to solve since there is no explicit dependence on the function that is being differentiated, i.e. dx/dt doesnt depend on x itself. The functions can be solved by multiplying through by dt and then integrating both sides....
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This note was uploaded on 01/20/2012 for the course PHYSICS 331 taught by Professor Staff during the Fall '09 term at Indiana.

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diff_eq - Electricity and Magnetism I (P331) M. R. Shepherd...

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