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Autonomous


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 t). A second-order autonomous differential equation is of the form F(y,y^',y^(''))=0, where y^'=dy/dt=v. By the chain rule, y^('') can be expressed as

 y^('')=v^'=(dv)/(dt)=(dv)/(dy)(dy)/(dt)=(dv)/(dy)v.

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 n first-order ODEs can be written as an autonomous system of n+1 ODEs by letting t=x_(n+1) and increasing the dimension of the system by 1 by adding the equation

 (dx_(n+1))/(dt)=1.

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Cite this as:

Weisstein, Eric W. "Autonomous." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Autonomous.html

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