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=== Higher-order process ===
While the Euler method integrates a first-order ODE, any ODE of order <math>N</math> can be represented as a system of first-order ODEs. When given the ODE of order <math>N</math> defined as

:<math> y^{(N+1)}(t) = f\left(t, y(t), y'(t), \ldots, y^{(N)}(t)\right) ,</math>

as well as <math>h</math>, <math>t_0</math>, and <math>y_0,y'_0,\dots, y^{(N)}_0</math>, we implement the following formula until we reach the approximation of the solution to the ODE at the desired time:

:<math> \vec{y}_{i+1}
  = \begin{pmatrix} y_{i+1}\\ y'_{ i+1 } \\ \vdots\\ y_{i+1}^{(N-1)}\\ y_{i+1}^{(N)} \end{pmatrix}
  = \begin{pmatrix} y_{i}+h\cdot y'_i\\ y' _{i}+h\cdot y''_{i}\\  \vdots\\ y^{(N-1)}_{i}+h\cdot y^{(N)}_{i}\\ y^{(N)}_{i}+h\cdot f\left(t_i, y_i, y'_i, \ldots, y^{(N)}_i\right) \end{pmatrix} </math>

These first-order systems can be handled by Euler's method or, in fact, by any other scheme for first-order systems.<ref>{{harvnb|Butcher|2003|p=3}}; {{harvnb|Hairer|N?rsett|Wanner|1993|p=2}}</ref>