Euler's method is one of the simplest numerical methods for approximating solutions of first-order initial-value problems
\[
y'=f(x,y),\quad y(x_0)=y_0.
\]
The idea behind this method is to find approximate values for the solution at equally spaced numbers
where $h$ is the step size. The differential equation tells us that the slope at $(x_0,y_0)$ is $y'=f(x_0,y_0)$, so the approximate value of the solution when $x=x_1$ is
$$y_1=y_0+h\,f(x_0,y_0)$$
as it is shown in Figure 1.
Figure 1: Approximate value of the solution.
Similarly, $$y_2=y_1+h\,f(x_1,y_1).$$
In general $$y_n=y_{n-1}+h\,f(x_{n-1},y_{n-1}).$$
Simulation
The following simulation shows exact solutions and approximate solutions of first-order initial-value problems $y'=f(x,y)$, $ y(x_0)=y_0$.
Things to try:
Right window
Change the function $f(x,y)$. Example: (x-2)(y^2-1)
Change the initial conditions $x_0,y_0$.
Change the value of the step size $h$.
Explore values $x_n,y_n$.
Activate the box Slope field.
Left window
Drag with mouse to change the axes view.
Zoom In/Out with mouse wheel.
Drag the x-axis (y-axis) with mouse, while pressing 'shift' key, to adjust aspect ratio
Sorry, the simulation is not supported for small screens.
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