>[!quote] In a Nutshell >Special category of [[ODEs - Ordinary Differential Equations|ordinary differential equations]] that does not depend on the independent variable. When considering $y=f(x)$ and therefore derivatives of $y$, th this means that $F\left(x, y, y', y'',\ \ldots,\ y^{(n)}\right) =F\left(y, y', y'',\ \ldots,\ y^{(n)}\right).$ >[!brainwaves] Why are they special ? >In physics, the autonomous variable $x$ is usually replaced by time $t$. Since the laws of nature are assumed to be unaffected by time, autonomous systems play a special role in many applications. --- #### Properties >[!success] Theorem >Solutions to autonomous ODEs are invariant under translations. If $y_1(x)$ is a solution, then $y_2(x)=y_1(x-x_0)$ with $x_0\in \mathbb{R}$ is also a solution. > >[!proof] Proof >Simply follows by replacing $x-x_0$ by another variable and verifying initial condition and unaffectedness of the original equation. > --- #### Solutions In general, any autonomous ODE or order $n$ can be reduced to an $n$-dimensional system of order $1$ (see [[ODEs - Ordinary Differential Equations|Theorem]]). It therefore suffices to consider solution techniques of one dimensional examples. However, there are only a handful of special cases that can always be solved explicitly: | General Form | Technique | Remarks | | -------------------------------------- | -------------------------------------------------------------------- | ------- | | $y^\prime=g(y)$ | [[Seperation of Variables for ODEs and PDEs]] | | | $y^{\prime \prime}=g(y, y^\prime)$ | Introduce new variable and reduce to first order. | | | $y^{\prime \prime}=g(y)$ | Can be rewritten using [[Seperation of Variables for ODEs and PDEs]] | | | $y^{\prime \prime}=(y^\prime)^{n}g(y)$ | | | Details on these techniques can e.g. be found on Wikipedia under https://en.wikipedia.org/wiki/Autonomous_system_(mathematics). Any autonomous equation of order $n \geq 3$ can only bes solved in exactly if they have general favorable properties such as linearity. >[!brainwaves] Intuition >This fact is not surprising considering that many [[Static, Dynamic and Stochastic Systems|dynamical systems]] (which are autonomous) in $3$ dimensions can be chaotic.