Question

[Differential Equations]: Intro to Systems

Determine a first-order system equivalent to the following equation (please see the equation in the comments).

Is the system autonomous? Assume the independent variable is t.

Answer 1

\[y ^{(4)}+\tan (y ^{'}+y ^{''})-y ^{'''}+e ^{y}=t ^{2}\]

Answer 2

If I'm not mistaken, I think an autonomous \(n\)-th order system is one that relates the \(n\)-th order derivative to a function of the dependent variable's \(n-1\) derivatives, so something like \(y^{(n)}=f(y^{(n-1)},y^{(n-2)},\ldots,y',y)\). Note the lack of an independent variable in this definition.

Answer 3

This page seems to agree, at least in the case of \(n=2\): http://mathworld.wolfram.com/Autonomous.html

Answer 4

Your answer makes sense, but I seem to have gotten that this is a non-autonomous system with the following conditions set:
\[u' _{1} = u _{2}\]\[u' _{2} = u _{3}\]\[u' _{3} = u _{4}\]\[u' _{4} = u _{5}\]\[u' _{5} = t ^{2} - \tan (u _{2} + u _{3}) + u _{4} - e ^{u _{1}}\]I assumed since the fifth and final expression relates to the variable t, that the system is non-autonomous.

Answer 5

Right, that's my point: the given system is *not* autonomous because it contains an expression with the independent variable.

Answer 6

...right. :P My bad, trying to teach myself this stuff has drained my brain of basic English comprehension. Thanks for your help!

Answer 7

You're welcome!