The Euler method is a numerical technique for approximating solutions to first-order differential equations. It's like taking baby steps along a curve, using the derivative to guide each step.
Consider a point $(x_n, y_n)$, where the derivative at this point is $$y'(x_n)=f(x_n,y_n)$$. The tangent line at this point is given by
$$y - y_n= y_n'(x_n, y_n)(x-x_n)$$
$$y = y_n + y_n'(x_n, y_n)(x-x_n)$$
Now let us consider a new point $x_{n+1}, y_{n+1}$, where $x_{n+1}$ is defined by $x_{n+1}= x_{n}+h$ for some small $h$.
Here comes the approximation step. We can imagine that, for some small $h$, the rate of change at the points $(x_n, y_n)$ and $(x_{n+1}, y_{n+1})$ is incredibly similar, to the point where they are effectively the same.
If the rate of change at these points are similar, the point $(x_{n+1}, y_{n+1})$ should lie on the tangent line of $(x_n, y_n)$.
Therefore, by substituting the point $(x_{n+1}, y_{n+1})$ into the tangent line:
$$y_{n+1} = y_n + y_n'(x_n,y_n)(x_{n+1}-x_n)$$
$$y_{n+1} = y_n + f(x_n,y_n)h$$
$$y_{n+1} = y_n + hy_n'(x_n)$$
Or alternatively, by letting $f(x_n, y_n) = y_n'(x_n,y_n)$
$$y_{n+1} = y_n + h\cdot f(x_n, y_n)$$
Where:
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