A differential equation or system of ordinary differential equations is said to be autonomous if it does not explicitly contain
the independent variable (usually denoted 
). A second-order autonomous differential equation is of
the form 
,
where 
. By the chain
rule, 
can be expressed as
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For an autonomous ODE, the solution is independent of the time at which the initial conditions are applied. This means
that all particles pass through a given point in phase space. A nonautonomous system
of 
first-order ODEs can be written as an
autonomous system of 
ODEs by letting 
and increasing the dimension of the system by 1 by adding the equation
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