When given the initial values for and , and the derivative of is given as a function of and denoted as , the Euler’s method is used to find from and :

Here, is a fixed step size, and . The value of is an approximation of the solution at , i.e., .

The Euler’s method is explicit, meaning the solution is an explicit function of for . Due to this characteristic, it is also referred to as the explicit Euler’s method or forward Euler’s method.

Generalization to Higher-order Process

While the Euler method is typically used to solve first-order ODEs, any ODE of order can be represented as a system of first-order ODEs. when given an ODE of order defined as along with a step size and initial conditions , we can use the following formula:

Variation

In addition to the explicit Euler method, there are two other variations of Euler methods that offer different trade-offs: