A rabbit starts at the origin and runs up the right branch of the parabola y=x^2 with speed a. At the same time a dog, running with speed b, starts at the point (c,0) and pursues the rabbit. Write a differential equation for the path of the dog
To find the path of the dog, we need to consider its position and velocity at any given time. Let's denote the position of the dog as (x_d, y_d) and the position of the rabbit as (x_r, y_r).
Since the rabbit is running along the right branch of the parabola y = x^2, its position can be described as (x_r, x_r^2). The velocity of the rabbit is given as the derivative of its position, which is (1, 2x_r).
The dog starts at the point (c, 0) and runs with a speed b. Hence, the magnitude of the dog's velocity is a constant b, and its direction is towards the rabbit. We can write the dog's velocity as (b, v_d), where v_d represents the dog's velocity in the y-direction.
To find the value of v_d, we need to consider the relative motion between the rabbit and the dog. The dog always runs towards the rabbit, so its velocity in the x-direction will be the difference between the velocities of the dog and rabbit in the x-direction. This can be written as v_d - 2x_r.
Finally, we can write the differential equation for the path of the dog by equating the velocities in the x and y directions:
dx_d/dt = b,
dy_d/dt = v_d - 2x_r.
Note that x_r and v_d are functions of time since they depend on the positions and velocities of the rabbit and dog at any given time.