forward euler method

[idle hollow]

bit confuse about the working of it

[dusky drift]

Do you mean the numeric method for ODEs?

So we're solving $\dot{x} = f(t, x)$ numerically

[fair carbon]

spindle

[dusky drift]

The idea is that we use Taylor series

$x(t) = x(t_0) + \dot{x}(t_0)(t - t_0) + O((t - t_0)^2)$

[fair carbon]

spindle

[dusky drift]

But we know from the equation that $\dot{x}(t_0) = f(t_0,x(t_0))$, so we can plug that into our equation

[fair carbon]

spindle

[dusky drift]

That's it

If you want to obtain terms of higher order, then just differentiate the ODE:
$$\ddot{x} = f_t(t,x) + f_x(t,x)\dot{x} = f_t(t,x) + f_x(t,x)f(t,x)$$

[fair carbon]

spindle

[dusky drift]

So now you can rewrite $x(t)$ as
$$x(t_0) + \dot{x}(t_0)(t-t_0) + \frac{1}{2}\ddot{x}(t_0)(t-t_0)^2 + O((t - t_0)^3)$$

[fair carbon]

spindle

[dusky drift]

And then plug in $\dot{x}(t_0)$, $\ddot{x}(t_0)$ using the equations