Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN: 9781133382119
Author: Swokowski
Publisher: Cengage
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[画像:An improved method that is similar to Euler's method is what is usually called the Improved Euler's method. It works like this: Consider an equation y' = f(x, y). From (xn, Yn), our approximation to the solution of the differential equation at the n-th stage, we find the next stage by computing the x-step Xn+1 = £n +h, and then k1, the slope at (xn, Yn). The predicted new value of the solution is Zn+1 = Yn +h. k1. Then we find the slope at the predicted new point k2 = f(xn+1, Zn+1) and get the corrected point by averaging slopes h Yn+1 = Yn + · (k1 + k2). 2 Suppose that we use the Improved Euler's method to approximate the solution to the differential equation dy dx 0.5y, y(0.5) = 9. = We let xo 0.5 and yo = = 9 and pick a step size h = 0.25. Complete the following table: n xn Yn k1 Zn+1 k2 0 0.59 -4 8-3.25 1 2 3 4 help (numbers) The exact solution can also be found for the linear equation. Write the answer as a function of x. y(x) = help (formulas) Thus the actual value of the function at the point x = 1.5 is y(1.5) = = help (numbers) While the approximate solution found above is y(1.5) ≈ Book: Section 1.7 of Notes on Diffy Qs help (numbers)]
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Transcribed Image Text:An improved method that is similar to Euler's method is what is usually called the Improved Euler's method. It works like this: Consider an equation y' = f(x, y). From (xn, Yn), our approximation to the solution of the differential equation at the n-th stage, we find the next stage by computing the x-step Xn+1 = £n +h, and then k1, the slope at (xn, Yn). The predicted new value of the solution is Zn+1 = Yn +h. k1. Then we find the slope at the predicted new point k2 = f(xn+1, Zn+1) and get the corrected point by averaging slopes h Yn+1 = Yn + · (k1 + k2). 2 Suppose that we use the Improved Euler's method to approximate the solution to the differential equation dy dx 0.5y, y(0.5) = 9. = We let xo 0.5 and yo = = 9 and pick a step size h = 0.25. Complete the following table: n xn Yn k1 Zn+1 k2 0 0.59 -4 8-3.25 1 2 3 4 help (numbers) The exact solution can also be found for the linear equation. Write the answer as a function of x. y(x) = help (formulas) Thus the actual value of the function at the point x = 1.5 is y(1.5) = = help (numbers) While the approximate solution found above is y(1.5) ≈ Book: Section 1.7 of Notes on Diffy Qs help (numbers)
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