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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)

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)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 92E
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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)
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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