07.03 Approximating Solutions Using Euler's Method

Find the missing values in the chart below: n xn yn dy/dx Δy = dy/dx • Δx 0 0 1 −1 0.5(-1) = -1/2 1 1/2 1/2 -1/4 0.5(0.25) = -1/8 2 A 3/8 1/8 0.5(0.125) = 1/16 3 3/2 7/16 5/16 0.5(0.3125) = 5/32 4 2 B

B. A = 1 and B = 19/32

Let y = f(x) be a solution to the differential equation dy/dx = x + y with the initial condition f(0) = m, where m is a constant. Using the initial condition and Δx = 1, Euler's Method gives the approximation f(2) ≈ 5. Find the value of m.

C. 1

Use Euler's Method with two equal steps to approximate y(1) to three decimal places given the differential equation dy/dx = e^xy and the initial condition y(0) = 1.

D. 2.559

How does the solution curve produced using Euler's Method for the differential equation dy/dx = f(x, y) compare with the actual curve when lim Δx (yn-1 + f(xn-1, yn-1)Δx)?

D. The solution curve mimics the actual curve perfectly.

Given the differential equation dy/dx = g(x, y) and initial condition g(0) = 1, Euler's Method produces the value y1 = 1 + h • g(0, 1), where h equals the step size. Find y2.

A. y2 = 1 + h • g(0, 1) + h • g(h, 1 + h • g(0, 1))