Euler's method is basically a way of approximating function values without going through the pain in the ass method of solving the differential equation. The equation used is pretty simple to understand.
y1 = y0 + Δy
where Δy = Δy/Δx * (Δx)
or as it is more commonly seen = dy/dx * (Δx)
Yup you guessed it! Delta x is the step, which is 1 here.
So we have y1 = y0 + slope * step
We start at 1 and move to 3 so we are going to have to do this twice
y1 = y0 + y'(1,2) (1) = 2 + (3/2)(1) = 3.5
So this point is (2 , 3.5)
Plug this in to find slope, again.
y2 = 3.5 + (12/7)(1) = 5.214
We are now at x=3
(3, 5.214)