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=== First-order process ===
When given the values for <math>t_0 </math> and <math>y(t_0) </math>, and the derivative of <math>y </math> is a given function of <math>t </math> and <math>y </math> denoted as <math>y'(t) = f\bigl(t,y(t)\bigr) </math>. Begin the process by setting <math>y_0=y(t_0) </math>. Next, choose a value <math>h</math> for the size of every step along t-axis, and set <math>t_n = t_0 + nh</math> (or equivalently <math>t_{n+1} = t_n + h</math>). Now, the Euler method is used to find <math>y_{n+1}</math> from <math>y_n</math> and <math>t_n</math>:<ref>{{harvnb|Butcher|2003|p=45}}; {{harvnb|Hairer|N?rsett|Wanner|1993|p=36}}</ref>

:<math> y_{n+1} = y_n + hf(t_n,y_n).</math>

The value of <math>y_n</math> is an approximation of the solution at time <math>t_n</math>, i.e., <math>y_n \approx y(t_n)</math>. The Euler method is [[explicit and implicit methods|explicit]], i.e. the solution <math>y_{n+1}</math> is an explicit function of <math>y_i</math> for <math>i \leq n</math>.