FirstOrderODEs_Primer - Primer on First-Order Ordinary...

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Primer on First-Order Ordinary Differential Equations ü Copyright Brian G. Higgins (2004) 1. Introduction In this notebook we present an introductory review of the properties and solutions of 1 st order ordinary differen- tial equations. In section 2 we introduce the concept of 1-parameter family of level curves which we show in later sections forms the building block for understanding the solution properties of 1 st order ODEs. We then review some basic definitions regarding ordinary differential equations, and then proceed with the main discussion on several methods for solving ODEs. The first method we consider is for linear differential equations which can be solved by finding an integrating factor. The next section introduces a solution method based on separation of variables that can be used to solve n on l inear 1 st order ODES. Each section has numerous worked out examples. These notes do not discuss exact ODEs or the use of homogeneous functions to solve first order ODES 2. Level Curves Concept 1: A function g H x, y L = c defines a one-parameter family of plane curves called level curves. Example 2.1: Level Curves Let T H x, y L define the temperature of a plane body. Then the function defined by T H x, y L = c (where c is a constant) represents a curve in the x - y plane of constant temperature called an isotherm . Each value of the constant c defines a different isotherm. Isotherms defined by T H x, y L = c are also called level curves. Example 2.2 : Mathematica experiments with level curves Suppose the family of level curves is given by (1) g H x, y L = y 2 + 3 xy + x 2 = c Display the level curves for c = 8 - 3, - 2, - 1, 0, 1, 2, 3 <
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Solution For each value of c we need to find the H x, y L pair that satisfies g H x, y L = y 2 + 3 xy + x 2 = c Since g H x, y L = c defines a surface of height c , we need to determine a contour plot of this function. This can be readily done in Mathematica using the ContourPlot function. Here is the code that achieves the desired result. ContourPlot @ y 2 + 3 xy + x 2 , 8 x, - 3, 3 < , 8 y, - 3, 3 < , ContourShading -> False, PlotPoints -> 100, ContourStyle -> RGBColor @ 0, 0, 1 DD ; -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 It is more instructive to plot particular level curves. For example, suppose we want the level curves for c = 8 - 3, - 2, - 1, 0, 1, 2, 3 < . This can be done in ContourPlot by specifying the option Contours. The result is shown below 2 FirstOrderODEs_Primer.nb
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ContourPlot @ y 2 + 3 xy + x 2 , 8 x, - 3, 3 < , 8 y, - 3, 3 < , Contours -> 8 - 3, - 2, - 1, 0, 1, 2, 3 < , ContourShading -> False, PlotPoints -> 100, ContourStyle -> RGBColor @ 0, 0, 1 DD ; -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Note that in this example the point H 0, 0 L lies at the intersection of two contours. It follows from the function g H x, y L that H 0, 0 L must lie on the contour for c = 0 . Indeed, one can show that for each value of c 0 there are two level curves that satisfy g H x, y L = c . We show this in the next plot for c = - 2 FirstOrderODEs_Primer.nb 3
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ContourPlot @ y 2 + 3 xy + x 2 , 8 x, - 3, 3 < , 8 y, - 3, 3 < , Contours -> 8 - 2 < , ContourShading -> False, PlotPoints -> 100, ContourStyle -> RGBColor @ 0, 0, 1 DD ; -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Thus when c = 0 , there is a an H x, y L pair, namely, H 0, 0 L that is common to both level curves. The implica-
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