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Unformatted text preview: Euler’s Method We have seen that few first order IVP’s can be solved explicitly, but that in many cases we can verify that a unique solution does exist. In practice, it will suffice to have an accurate approximation to that solution. Here we will examine one of the simplest (hence least accurate) methods, as an introduction to numerical techniques. We will again assume we have an IVP of the form y = f ( x,y ) , y ( x ) = y . Approximating a curve by line segments Suppose y ( x ) is the solution to the above IVP and we wish to approximate on the interval [ x ,x + α ] . For 1 ≤ k ≤ N , let h = α/N and define x k = x + kh. Then the points x ,x 1 ,...,x N divide the interval [ x ,x + α ] into N equal parts. If we knew the value of y ( x k ) for each k , we could make a reasonable approximation to y ( x ) by joining the points ( x k ,y ( x k )) by line segments. Call the approximating function ˆ y ( x ) . This is illustrated below in the case of N = 3 ....
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This note was uploaded on 03/03/2012 for the course MATH 4430 at Colorado.