Question

Use Euler's method with a step size of 0.1 to estimate (0.5)y(0.5), where ()y(x) is the solution of the initial-value problem y ′=9−y−xy, (0)=1y(0)=1.

a. 0.48
b. 0.53
c. 0.58
d. 0.63

Answer 1

To estimate (0.5)y(0.5) using Euler's method with a step size of 0.1, we need to find the derivative of y(x) and iterate through the process of approximating y at different points until we reach our desired value.

Step-by-step explanation:

To estimate (0.5)y(0.5) using Euler's method with a step size of 0.1, we can follow these steps:

1. First, we need to find the derivative of y(x) using the given initial-value problem: y ′=9−y−xy.
2. Next, we choose a starting point, which is (0,1) in this case. Using the derivative, we can calculate the approximate value of y at x=0.1.
3. We repeat step 2 with the new value of y and continue this process until we reach x=0.5.
4. Once we have the approximation for y at x=0.5, we can multiply it by 0.5 to get the estimate for (0.5)y(0.5).

By following these steps, we can obtain the estimated value. In this case, the estimated value will be one of the options a, b, c, or d provided in the question.