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.

To use Euler's method with a step size of 0.5, we can approximate the y-values of the solution of the initial-value problem y' = y - 5x, y(3) = 0.

Step 1: Determine the number of steps Since the step size is 0.5, we need to take 4 steps to compute y1, y2, y3, and y4.

Step 2: Initialize the variables Let's start by setting x0 = 3 and y0 = 0, which are the initial values given in the problem.

Step 3: Compute the approximate y-values Using Euler's method, we can compute the approximate y-values as follows:

For i = 1: x1 = x0 + h = 3 + 0.5 = 3.5 y1 = y0 + h * (y'0) = 0 + 0.5 * (y0 - 5x0) = 0 + 0.5 * (0 - 5*3) = -7.5

For i = 2: x2 = x1 + h = 3.5 + 0.5 = 4 y2 = y1 + h * (y'1) = -7.5 + 0.5 * (y1 - 5x1) = -7.5 + 0.5 * (-7.5 - 5*3.5) = -18.25

For i = 3: x3 = x2 + h = 4 + 0.5 = 4.5 y3 = y2 + h * (y'2) = -18.25 + 0.5 * (y2 - 5x2) = -18.25 + 0.5 * (-18.25 - 5*4) = -34.625

For i = 4: x4 = x3 + h = 4.5 + 0.5 = 5 y4 = y3 + h * (y'3) = -34.625 + 0.5 * (y3 - 5x3) = -34.625 + 0.5 * (-34.625 - 5*4.5) = -57.3125

Therefore, the approximate y-values are: y1 ≈ -7.5 y2 ≈ -18.25 y3 ≈ -34.625 y4 ≈ -57.3125