Question

dy/dt=αy^(2/3)-βy is this differential equation linear or nonlinear? autonomous or non-autonomous? Please explain how you know. Thanks

Answer 1

@dan815

Answer 2

A linear first order differential equation is of the form
\[f(t)\frac{dy}{dt}+g(t)y=h(t)\]
Notice how \(f,g,h\) are functions of \(t\) only? and that the highest power of \(y\) and its derivative is one?

Answer 3

I should be more precise: the ONLY power of \(y\) is the first power.

As for autonomy, an autonomous equation is an equation that doesn't explicitly depend on the independent variable, which in this case is \(t\). This means an autonomous equation would be of the form
\[f_n(y)\frac{d^ny}{dt^n}+\cdots+f_1(y)\frac{dy}{dt}+f_0(y)=g(y)\]

Answer 4

@SithsAndGiggles , so it isn't autonomous, correct?

Answer 5

No, it is autonomous. One way to think of it (in regards to first order equations) is that an autonomous equation can be separated such that you can write it as
\[\frac{dy}{dt}=\frac{g(t)}{f(y)}~~\iff~~f(y)~dy=g(t)~dt\]
where \(g(t)=1\).

Answer 6

(Sorry, heh.)

Could you give me an example that has similar form to an Autonomous DE but is non-autonomous?