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\end{align}</math>
where Δ''t'' is the time step and ''t<sub>n</sub>'' = ''t''<sub>0</sub>
The difference with the standard Euler method is that the semi-implicit Euler method uses ''v''<sub>''n''+1</sub> in the equation for ''x''<sub>''n''+1</sub>, while the Euler method uses ''v<sub>n</sub>''.
Applying the method with negative time step to the computation of <math>(x_n, v_n)</math> from <math>(x_{n+1}, v_{n+1})</math> and rearranging leads to the second variant of the semi-implicit Euler method
:<math>\begin{align}
x_{n+1} &= x_n + f(t_n, v_n) \, \Delta t\\[0.3em]
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