I have the differential equation:

$\displaystyle x' = 1 + x^2$, where $\displaystyle x(0) = 1$

which I have to solve numerically. The problem is just that I don't understand how the method works, and it's hard to find any good explanations online.

I would really appriciate it if someone explained how to start, like how do you find $\displaystyle x_1$?

The equation is to be solved on the intevall [0; 0.6] using 6 steps, so that the steplenght is 0.1