MHB Understanding Autonomous Functions: Helping Me Grasp It!

[simo1]

I do not think I understand fully the concept of determining whether a function is autonomous> may you please help me understnad
eg I was given thsi function
x' = x^3 x(1)=1
i said

f(z)=z^3
where f'(z) = 3x^2 and the domain of f is in all R where the domain of f is also in all R hence f' is contionous then it is autonous. but I don't even know why we do this

 

[Fantini]

Re: autonomous DE

Examples to compare may help: $$\begin{cases} x' = x^3 & \text{is autonomous,} \\ x' = t x^3 & \text{is not.} \end{cases}$$ In the second case we have an explicit dependence on the independent variable in the differential equation.