Ordinary differential equation

Ordinary differential equation (ODE) is a differential equation which consists only one variable. (it makes it different from Partial differential equation). Therefore, ODE is an equation that can be presented like this

                     $F(x,y,{d \over dx}y,{d^{2} \over dx^{2}}y,...,{d^{n} \over dx^{n}}y)=0$           $(1)$

where $y(x)$ is unknown function that depends on $x$. $y'(x)$ means a differentiation by $x$.

undependent variable $x$ is often interpreted as time (especially in differential equations of Mathematical physics). variable $y$ is a certain value or a set of some values(if $y$ is a Vector-valued function).

the most common type of differential equations looks like this

$y^{n}=F(x,y,y',y'',...,y^{i})$ $(2)$

where $i<n$.

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