Use Euler's method with step size 0.5 to compute the approximate y-values y1, y2, y3 and y4 of the solution of the initial-value problem y' = y − 5x, y(3) = 0.y1 = y2 = y3 = y4 =
Question
Use Euler's method with step size 0.5 to compute the approximate y-values y1, y2, y3 and y4 of the solution of the initial-value problem y' = y − 5x, y(3) = 0.y1 = y2 = y3 = y4 =
Solution
To use Euler's method with a step size of 0.5, we need to follow these steps:
Step 1: Determine the number of iterations. In this case, we need to compute y1, y2, y3, and y4, so we will have four iterations.
Step 2: Set up the initial conditions. The initial condition given is y(3) = 0. Therefore, we start with x0 = 3 and y0 = 0.
Step 3: Calculate the approximate y-values using Euler's method. The formula for Euler's method is:
yn+1 = yn + h * f(xn, yn)
where yn+1 is the next y-value, yn is the current y-value, xn is the current x-value, h is the step size, and f(xn, yn) is the derivative of y with respect to x evaluated at xn and yn.
Using the given initial-value problem, we have f(x, y) = y - 5x.
For the first iteration (n = 1): x1 = x0 + h = 3 + 0.5 = 3.5 y1 = y0 + h * f(x0, y0) = 0 + 0.5 * (0 - 5 * 3) = -7.5
For the second iteration (n = 2): x2 = x1 + h = 3.5 + 0.5 = 4 y2 = y1 + h * f(x1, y1) = -7.5 + 0.5 * (-7.5 - 5 * 3.5) = -16.25
For the third iteration (n = 3): x3 = x2 + h = 4 + 0.5 = 4.5 y3 = y2 + h * f(x2, y2) = -16.25 + 0.5 * (-16.25 - 5 * 4) = -27.125
For the fourth iteration (n = 4): x4 = x3 + h = 4.5 + 0.5 = 5 y4 = y3 + h * f(x3, y3) = -27.125 + 0.5 * (-27.125 - 5 * 4.5) = -40.5625
Therefore, the approximate y-values are: y1 = -7.5 y2 = -16.25 y3 = -27.125 y4 = -40.5625
Similar Questions
Use Euler's method with step size 0.2 to estimate y(1.4), where y(x) is the solution of the initial-value problem y' = 4x − 3xy, y(1) = 0. (Round your answer to four decimal places.)y(1.4) =
Use Euler's method with step size 0.1 to estimate y(2.5), where y(x) is the solution of the initial-value problem y' = 3y + 2xy, y(2) = 1. (Round your answer to four decimal places.)
Consider the initial value problemy′(t) = cos(y), y(0) = 0.(i) Apply 4 steps of forward Euler method with the step size h = 12 to approximately findy(2). Show at least 4 decimal places in your calculations. [10 marks](ii) Apply 2 steps of Heun’s method with the step size h = 1 to approximately find y(2).Show at least 4 decimal places in your calculations.
Let y = f(x) be the solution to the differential equation 𝑑𝑦𝑑𝑥=𝑓′(𝑥) with initial condition f(2) = 3. Selected values of f' are given in the table above. What is the approximation for f(2.4) if Euler’s method is used, starting at x = 2 with two steps of equal size?
Find the solution to the initial value problemy′−e−t+5y=−5, y(0)=−1.𝑦′−𝑒−𝑡+5𝑦=−5, 𝑦(0)=−1.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.