euler method for simple motion

Solving an ODE for simple motion using Euler method

intro

Euler method, which is also called forward Euler method, is a first-order numerical procedure for solving ordinary differential equation (ODE) as initial value problem (IVP) ¹. This method can be described in only five lines of pseudocode using variables with index ² or without index ³. And it is an an explicit method for solving IVP ⁴. The use of Euler method for some simple motions is discussed here.

ode

An ODE in the form of

$$\tag{1} \frac{dx(t)}{dt} = f(t, x(t)) $$

must be supplied with initial value

$$\tag{2} x(t_0) = x_0 $$

to be an IVP.

solution

With time step $\Delta t$, Eqn (1) will have

$$\tag{3} x(t + \Delta t) = x(t) + \Delta t f(t, x(t)). $$

as an iterative solution with Eqn (2) as the initial value. It is convenient to use indexed variables for Eqn (2) as follow

$$\tag{4} x_{n+1} = x_n + h f(t_n, x_n), $$

where $h = \Delta t$, $t_n = t_0 + nh$, $x_n = x(t_n)$.

notes