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Unformatted text preview: Chapter 13 Ordinary Differential Equations Mathematical models in many different fields. Systems of differential equations form the basis of mathematical models in a wide range of fields from engineering and physical sciences to finance and biological sciences. Differential equations are relations between unknown functions and their derivatives. Computing numerical solutions to differential equations is one of the most important tasks in technical computing, and one of the strengths of Matlab . If you have studied calculus, you have learned a kind of mechanical process for differentiating functions represented by formulas involving powers, trig functions, and the like. You know that the derivative of x 3 is 3 x 2 and you may remember that the derivative of tan x is 1 + tan 2 x . That kind of differentiation is important and useful, but not our primary focus here. We are interested in situations where the functions are not known and cannot be represented by simple formulas. We will compute numerical approximations to the values of a function at enough points to print a table or plot a graph. Imagine you are travelling on a mountain road. Your altitude varies as you travel. The altitude can be regarded as a function of time, or as a function of longi- tude and latitude, or as a function of the distance you have traveled. Lets consider the latter. Let x denote the distance traveled and y = y ( x ) denote the altitude. If you happen to be carrying an altimeter with you, or you have a deluxe GPS system, you can collect enough values to plot a graph of altitude versus distance, like the first plot in figure 13.1. Suppose you see a sign saying that you are on a 6% uphill grade. For some Copyright c 2009 Cleve Moler Matlab R is a registered trademark of The MathWorks, Inc. TM August 8, 2009 1 2 Chapter 13. Ordinary Differential Equations 6000 6010 6020 6030 6040 6050 6060 6070 950 1000 1050 1100 1150 altitude 6000 6010 6020 6030 6040 6050 6060 6070-20-10 10 20 slope distance Figure 13.1. Altitude along a mountain road, and derivative of that alti- tude. The derivative is zero at the local maxima and minima of the altitude. value of x near the sign, and for h = 100, you will have y ( x + h )- y ( x ) h = . 06 The quotient on the left is the slope of the road between x and x + h . Now imagine that you had signs every few meters telling you the grade at those points. These signs would provide approximate values of the rate of change of altitude with respect to distance travelled, This is the derivative dy/dx . You could plot a graph of dy/dx , like the second plot in figure 13.1, even though you do not have closed-form formulas for either the altitude or its derivative. This is how Matlab solves differential equations. Note that the derivative is positive where the altitude is increasing, negative where it is decreasing, zero at the local maxima and minima, and near zero on the flat stretches....
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