This routine is used to determine if a system of differential equations, or a single differential equation, is strictly autonomous.  "Strictly" means that the independent variable does not appear as a part or argument of any terms, other than in calls to 'diff' or 'D', or when explicitly defined as part of the dependent variable.

'true' is returned if the system is determined to be autonomous by the above criteria; otherwise, 'false' is returned.

There can be only one independent variable.

This function is part of the DEtools package, and so it can be used in the form autonomous(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[autonomous](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{DE}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=y\left(x\right)\sin \left(y\left(x\right)-\cos \left(\sin \left(\tan \left(\mathrm{Bessel}\left(y\left(x\right)-{y\left(x\right)}^{\frac{1}{2}}\,3\right)\right)\right)\right)\right)$

$\mathrm{DE}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=-y\left(x\right)\sin \left(-y\left(x\right)+\cos \left(\sin \left(\tan \left(\mathrm{Bessel}\left(y\left(x\right)-\sqrt{y\left(x\right)}\,3\right)\right)\right)\right)\right)$

$\mathrm{autonomous}\left(\mathrm{DE}\,y\left(x\right)\,x\right)$

$\mathrm{true}$

$\mathrm{autonomous}\left(\sin \left(z\left(t\right)-{z\left(t\right)}^2\right){\mathrm{D}}^{\left(4\right)}\left(z\right)\left(t\right)-\cos \left(z\left(t\right)\right)-5\,z\,t\right)$

$\mathrm{true}$

$\mathrm{DE2}≔\mathrm{diff}\left(x\left(s\right)\,s\right)-x\left(s\right)\cos \left(\arctan \left(x\left(s\right)\right)\right)=\arctan \left(s\right)$

$\mathrm{DE2}≔\frac{\ⅆ}{\ⅆs}x\left(s\right)-\frac{x\left(s\right)}{\sqrt{1+{x\left(s\right)}^2}}=\arctan \left(s\right)$

$\mathrm{autonomous}\left(\mathrm{DE2}\,\left\{x\right\}\,s\right)$

$\mathrm{false}$

$\mathrm{DE3}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=\sin \left(y\left(x\right)\right)z\left(x\right),\mathrm{diff}\left(z\left(x\right)\,x\right)\cos \left(z\left(x\right)\right)=y\left(x\right)\mathrm{Bessel}\left(y\left(x\right)\,3\right)$

$\mathrm{DE3}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\sin \left(y\left(x\right)\right)z\left(x\right),\left(\frac{\ⅆ}{\ⅆx}z\left(x\right)\right)\cos \left(z\left(x\right)\right)=y\left(x\right)\mathrm{Bessel}\left(y\left(x\right)\,3\right)$

$\mathrm{autonomous}\left(\left\{\mathrm{DE3}\right\}\,\left\{y\,z\right\}\,x\right)$

$\mathrm{true}$