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Unformatted text preview: SECTION 2.7 SOME NUMERICAL STUFF: EULER'S METHOD Suppose we have a first order differential equation y (t) = f (t, y) with given initial value y(t0 ) = y0 . Further suppose that f (t, y) is really nice BUT that the equation is not linear, not separable, not exact, not anything that would allow us to find an analytic solution. However, to solve our engineering/physics/musical/whatever problem we really need values for y(t) at various t's. What can we do? EXAMPLE. Suppose we know that the quantity y(t) satisfies y = y(3 - ty) and that y(0) = 1. Our problem needs the values y(0.2), y(0.4), and y(0.6). What, exactly, does f (t, y), that is, y(3 - ty) in the example, tell us? How can we use this to produce approximate values of y(t)? EXAMPLE. Suppose we know that the quantity y(t) satisfies y = y(3 - ty) and that y(0) = 1. Use Euler's Method with t = h = 0.1 to approximate the values y(0.2), y(0.4), and y(0.6). Several of your homework problems are sort of fake, in that you could actually find analytical solutions for y(t). You could then compare the values produced by Euler's Method with the exact values for y(t). There is much much more to be said about these ideas, for example, all of Chapter 8, but we leave it here. HOMEWORK: SECTION 2.7 ...
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