Can an autonomous differential equation be nonhomogeneous?

Question

When a differential equation $\dfrac{dy}{dt}=f(y)$ does not depends on $y$ then it is autonomous, and it can't be non-homogeneous?

Can anyone explain this? Thanks!

Answer 1

$\frac{dy}{dt}=f(y)$ is an autonomous differential equation irrespective of whether $f$ depends on $y$. The only condition for autonomy is that $f$ should not be a function of $t$.

If $f$ does not depend even on $y$, then it is a constant function.

There are two definitions of homogeneous.

1. A first-order ODE is said to be homogeneous if it can be written in the form$$\frac{dy}{dt}=g\left(\frac yt\right)$$If you use this definition, then $\frac{dy}{dt}=k\in\mathbb R$ is homogeneous. Other examples include $$\frac{dy}{dt}=\frac{y^2+t^2}{yt},~\frac{dy}{dt}=\frac{y^3+ty^2}{t^3}~\text{ etc.}$$
2. A linear ODE is also said to be homogeneous if it has no term independent of the dependent variable or its derivatives with respect to the independent variable. Since $f$ is independent of $y$, $\frac{dy}{dt}=k\in\mathbb R$ is not homogenous using this definition.

Answer 2

$\frac{dy}{dt}=f(y)$ does not depend on $t$ !!

The terms "homogeneous" and "non-homogeneous" only make sense in connection with linear differential equations.

In general,

$$\frac{dy}{dt}=f(y)$$

is not linear. For example $\frac{dy}{dt}=y^2$ or $\frac{dy}{dt}=\sin (y),.....$