at time , i.e., . The Euler method is explicit, i.e. the solution is an explicit function of for . Higher-order process While the Euler method integrates a first-order ODE, any ODE of order can be represented as a system of first-order ODEs. When given the ODE of order defined as we implement the following formula until we reach the approximation of the solution to the ODE at the desired time: These first-order systems can be handled by Euler's method or, in fact, by any other scheme for first-order systems. Example Given the initial value problem we would like to use the Euler method to approximate . Using step size equal to 1 () The Euler method is so first we must compute . In this simple differential equation, the function is defined by . We have By doing the above step, we have found the slope of the line that is tangent to the solution curve at the point . Recall that the slope is defined as the change in divided by the change in , or . The next step is to multiply the above value by the step size , which we take equal to one here: Since the step size is the change in , when we multiply the step size and the slope of the tangent, we get a change in value. This value is then added to the initial value to obtain the next value to be used for computations. The above steps should be repeated to find , and . Due to the repetitive nature of this algorithm, it can be helpful to organize computations in a chart form, as seen below, to avoid making errors. {| class="wikitable" |- ! !! !! !! !! !! !! |- | 0 || 1 || 0 || 1 || 1 || 1 || 2 |- | 1 || 2 || 1 || 2 || 1 || 2 || 4 |- | 2 || 4 || 2 || 4 || 1 || 4 || 8 |- | 3 || 8 || 3 || 8 || 1 || 8 || 16 |} The conclusion of this computation is that . The exact solution of the differential equation is , so . Although the approximation of the Euler method was not very precise in this specific case, particularly due to a large value step size , its behaviour is