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**Unformatted text preview: **9 Euler’s Numerical Method In the last chapter, we saw that a computer can easily generate a slope field for a given first-order differential equation. Using that slope field we can sketch a fair approximation to the graph of the solution y to a given initial-value problem, and then, from that graph, we find find an approximation to y ( x ) for any desired x in the region of the sketched slope field. The obvious question now arises: Why not let the computer do all the work and just tell us the approximate value of y ( x ) for the desired x ? Well, why not? In this chapter, we will develop, use, and analyze one method for generating a “numerical solution” to a first-order differential equation. This type of “solution” is not a formula or equation for the actual solution y ( x ) , but two lists of numbers, { x , x 1 , x 2 , x 3 , . . . , x N } and { y , y 1 , y 2 , y 3 , . . . , y N } with each y k approximating the value of y ( x k ) . Obviously, a nice formula or equation for y ( x ) would be usually be preferred over a list of approximate values, but, when obtaining that nice formula or equation is not practical, a numerical solution is better than nothing. The method we will study in this chapter is “Euler’s method”. It is but one of many methods for generating numerical solutions to differential equations. We choose it as the first numerical method to study because is relatively simple, and, using it, you will be able to see many of the advantages and the disadvantages of numerical solutions. Besides, most of the other methods that might be discussed are refinements of Euler’s method, so we might as well learn this method first. 9.1 Deriving the Steps of the Method Euler’s method is based on approximating the graph of a solution y ( x ) with a sequence of tangent line approximations computed sequentially, in “steps”. Our first task, then, is to derive a useful formula for the tangent line approximation in each step. 191 192 Euler’s Numerical Method (a) (b) X X Y Y y ( x ) L k x k Delta1 x x k + Delta1 x Delta1 y y ( x k ) y ( x k ) + Delta1 y y ( x k + Delta1 x ) x y ( x 1 , y 1 ) L 1 ( x 2 , y 2 ) L 2 ( x 3 , y 3 ) L 3 ( x 4 , y 4 ) L 4 ( x 5 , y 5 ) L 5 Figure 9.1: (a) A single tangent line approximation for the Euler method, and (b) the approximation of the solution curve generated by five steps of Euler’s method. The Basic Step Approximation Let y = y ( x ) be the desired solution to some first-order differential equation dy dx = f ( x , y ) , and let x k be some value for x on the interval of interest. As illustrated in figure 9.1a, ( x k , y ( x k )) is a point on the graph of y = y ( x ) , and the nearby points on this graph can be approximated by corresponding points on the straight line tangent at point ( x k , y ( x k )) (line L k in figure 9.1a)....

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