Please tell whether im correct. decide whether function is autonomous ODE

1)dy/dx=(y-1)/x, i think this is autonomous since it doesnt depend on x right?

2)dy/dx=y-x, this is not since it depends on x

To determine whether a differential equation is autonomous, you need to check if it depends on the independent variable, which in this case is typically denoted by "t" instead of "x".

1) In the equation dy/dx = (y-1)/x, since the equation is in terms of "x" and not "t", it appears as if it doesn't depend on the independent variable. However, we need to be careful because sometimes equations can be rewritten in terms of a different variable without changing their nature. To confirm whether it is autonomous, we should substitute "y" for a function of "t" (let's say "y(t)") and "x" as an independent variable. This way, the equation becomes dy/dt = (y(t)-1)/x. Since this still depends on "x", we can conclude that the equation is not autonomous.

2) For the equation dy/dx = y - x, it is clear that it depends on the independent variable "x". So, your understanding is correct, and this equation is not autonomous.