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 the equation depends explicitly on the independent variable (in this case, x).
Let's analyze the given equations:
1) dy/dx = (y-1)/x:
In this equation, there is no explicit dependence on x. The presence of (y-1)/x doesn't change based on the value of x itself. The equation only involves the dependent variable y. Therefore, this equation is indeed autonomous.
2) dy/dx = y - x:
In this equation, the variable x is explicitly present and affects the equation regardless of the value of y. Hence, this equation is not autonomous.
Your analysis is correct. The first equation is autonomous because it does not depend explicitly on the variable x, while the second equation is not autonomous since it does depend on x.