Differential Equation of Parabolas with Directrices Parallel to x-axis

To derive the differential equation for all parabolas whose directrices are parallel to the x-axis, we can use the standard equation of a parabola with directrix at y = k:

(y - k)^2 = 4p(x - h)

where (h, k) is the vertex and p is the distance from the vertex to the focus.

Since the directrix is parallel to the x-axis, we have k = -p. Substituting this into the equation, we get:

(y + p)^2 = 4p(x - h)

Differentiating both sides with respect to x, we get:

2(y + p) (dy/dx) = 4p

Simplifying, we get:

dy/dx = 2p/(y + p)

To eliminate p, we can use the fact that the distance from the vertex to the focus is equal to the distance from the vertex to the directrix. Since the directrix is parallel to the x-axis, this distance is simply the absolute value of p. Therefore, we have:

p = |(y - k)/2|

Substituting this into the previous equation, we get:

dy/dx = 2|(y - k)/2|/(y + |(y - k)/2|)

Simplifying, we get:

dy/dx = (y - k)/|y - k|

Taking the derivative of both sides with respect to x, we get:

d^2y/dx^2 = d/dx[(y - k)/|y - k|]

Using the quotient rule, we get:

d^2y/dx^2 = [(dy/dx)|y - k| - (y - k) d/dx|y - k|]/|y - k|^2

Since d/dx|y - k| = 0 when y = k, we have:

d^2y/dx^2 = [(dy/dx)|y - k|]/|y - k|^2

To eliminate the absolute value, we can consider two cases:

Case 1: y > k

In this case, we have |y - k| = y - k. Substituting this into the previous equation, we get:

d^2y/dx^2 = [(dy/dx)(y - k)]/(y - k)^2

Simplifying, we get:

d^2y/dx^2 = dy/dx/(y - k)

Taking the derivative of both sides with respect to x, we get:

d^3y/dx^3 = d/dx[dy/dx/(y - k)]

Using the quotient rule, we get:

d^3y/dx^3 = [(d^2y/dx^2)(y - k) - dy/dx]/(y - k)^2

Substituting d^2y/dx^2 = dy/dx/(y - k), we get:

d^3y/dx^3 = -2(dy/dx)/(y - k)^3

Case 2: y < />

In this case, we have |y - k| = -(y - k). Substituting this into the previous equation, we get:

d^2y/dx^2 = [(dy/d