Use Euler's method with step size 0.1 to estimate y(0.5), where y(x) is the solution of the initial-value problem y ' = y + 4xy, y(0) = 1. (Round the answer to four decimal places.)
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To use Euler's method to estimate y(0.5) for the given initial-value problem, we need to perform the following steps:
1. Start by setting up the step size, h. In this case, the step size is given as 0.1.
2. Define the initial condition. We are given that y(0) = 1.
3. Set up a table to calculate the values of y(x) at each step. The table should have two columns, one for x and one for y.
4. Begin the calculation by setting x = 0 and y = 1 (using the initial condition).
5. Calculate the slope at the current values of x and y. The slope, dy/dx, is given by y' = y + 4xy.
In this case, dy/dx = y + 4xy = 1 + 4(0)(1) = 1.
6. Use the slope to calculate the change in y over the step size. The change in y, dy, is given by dy = dy/dx * h.
In this case, dy = 1 * 0.1 = 0.1.
7. Update the values of x and y by adding the step size and the change in y, respectively. x_new = x + h and y_new = y + dy.
In this case, x_new = 0 + 0.1 = 0.1 and y_new = 1 + 0.1 = 1.1.
8. Record the values of x and y in the table.
9. Repeat steps 5 to 8 until you reach the desired value of x, which in this case is x = 0.5.
Perform the calculation four more times, each time using the updated values of x and y to calculate the new slope, change in y, and updated values of x and y.
10. Once you have reached x = 0.5, round the final value of y to four decimal places to get the estimated value of y(0.5).
By following these steps, you should be able to use Euler's method to estimate y(0.5) for the given initial-value problem.