# Differential Equations

A **differential equation system** is given by: for all *t* ∈
* * *Reals*_{+}

*z'*(* t*)

*=*

*g*(

*z*(

*t*),

*v*(

*t*)).

Here

z'() is the derivative oftzwith respect tot, evaluated att;

t∈Reals_{+}stands for continuous time;

z:Reals_{+}→Realsis the state response; and^{N}

v:Reals_{+}→Realsis the input signal.^{N}

Here, the state is updated continuously rather than discretely, but otherwise, this is similar to a state machine with an infinite state space. In fact, we can approximate the derivative by the difference

*z*(*t* + δ ) - *z*(*t*)
≈
δ
*g*(*z*(*t*), *v*(*t*)).

If we sample *z *and *v *at times 0, δ , 2δ , **...**, *nδ*, (*n*+1)δ , **... **and denote the *n*th sample value by

*s*(*n*) = *z*(*n*δ*
*), *x*(*n*) = *v*(*n*δ ),

we get the **difference equation**

*s*(*n* + 1) - *s*(*n*) = δ
*g*(*s*(*n*), *x*(*n*))

This is called a **difference equation** because the difference
on the left is analogous to a differential in a differential equation.
This can be rewritten as a state update equation,

*s*(*n* + 1) = *s*(*n*) + δ
*g*(*s*(*n*), *x*(*n*))

Thus, a discrete approximation to a differential equation is a state machine with an infinite state space.