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 − 3x, y(2) = 0.

  see below

Step-by-step explanation:

For problems of this nature, it is convenient to use a spreadsheet to do the calculations. At each stage, the next value of y is ...

  y[n] = y'·Δx + y[n-1] . . . . where ∆x is the step size, and y[n-1] is the previous value of y, the same value that goes into the calculation of y'.

Of course, y' is given by the equation in the problem statement.

We are given ...

  y0 = 0 . . . for x=2

Then ...

  y1 = (0 -3(2))·0.5 + 0 = -3

  y2 = (-3 -3(2.5))·0.5 + (-3) = -8.25

and so it goes.

The numbers listed in the "y" column in the attached spreadsheet are y0 through y4 rounded to 2 decimal places. (Calculations are carried to full precision.)

_____

Calculation Simplified

For the given step size, the calculation can be simplified to ...

  y[n] = 1.5(y[n-1] -x[n-1])

so, ...

  y3 = 1.5(-8.25 -3.0) = -16.875

  y4 = 1.5(-16.875 -3.5) = -30.5625

_____

Actual Solution

The actual solution to this differential equation is ...

  y = 3(1+x) -9e^(x-2)

The spreadsheet also shows the actual value of y and the percentage error in the estimate with the given step size. Changing the step size to 0.02 reduces the error to less than 3% over the same x-interval [0, 4]. Of course, many more steps are required.