Chapter 8.3, Problem 21ES

A slope field of the form $y^{\prime }=f\left(y\right)$ is said to be autonomous.

(a) Explain why the tangent segments along any horizontal line will be parallel for an autonomous slope field.

(b) The word autonomous means "independent?' In what sense is an autonomous slope field independent?

(c) Suppose that $G\left(y\right)$ is an antiderivative of $1/\left[f\left(y\right)\right]$ and that C is a constant. Explain why any differentiable function defined implicitly by $G\left(y\right)-x=C$ will be a solution to the equation $y^{\prime }=f\left(y\right).$