Euler, without correction, is the simplest possible method of solution approximation. Are you SURE you are serious about this class? Really, if you are struggling with this, you will not be a happy camper at all in the real near future.

Just step through it.

dy/dx = 1 + xy

y(1) = 0

Then dy/dx around (1,0) is approximately 1 + (1)(0) = 1

y(1.1) = 0 + 0.1*1 = 0.1

The dy/dx around (1.1,0.1) = 1 + 1.1*0.1 = 1 + 0.11 = 1.11

y(1.2) = 0.1 + 0.1*1.11 = 0.211

The dy/dx around (1.2,0.211) = 1 + 1.2*0.211 = 1 + 0.12532 = 1.12532

Two things questionable with this presentation:

1) It was not necessary to calculate the last value. We were interested in y(1.2).

2) I carried all decimal places in all calculations. One may wish to concern oneself with proper significant figures. However, it pays to pay attention to all kinds of error.