The phrase "y is proportional to x" implies that y is related to x via the equat...

The phrase "y is proportional to x" implies that y is related to x via the equation y = kx, where k is a constant. In a similar manner, "y is proportional to the square of x" implies ,"y is proportional to the product of x and z" implies y = kxz, and "y is inversely proportional to the cube of f implies . For example, when Newton proposed that the force of attraction of one body on another is proportional to the

product of the masses and inversely proportional to the square of the distance between them, we can immediately write

where G is the constant of proportionality, usually known as the universal gravitational constant. In Exercise, use these ideas to model each application with a differential equation. All rates are assumed to be with respect to time.

Use Newton's law to develop the equation of motion for the particle in Exercise 8 if the force is proportional to, but opposite the square of the particle's velocity.

Reference: Exercise 8:

A particle moves along the x-axis, its position from the origin at time t given by x (t). A single force acts on the particle that is proportional to, but opposite the object's displacement. Use Newton's law to derive a differential equation for the object's motion.