Parabola with vertex and focus on the x axis.

Let’s consider the parabola’s equation which is;

And from that equation, let’s consider that (h,0) is the parabola’s vertex and the focus of each vertex is (h+p,0). And why does the vertex represent as (h,0) and the focus is (h+p,0). It is because the vertex and the focus is said to be changing its location on the x axis only.

From the condition of the said family of parabola, we come up to its fixed equation whish is:

*where h is any point along the x axis and the variable p is the distance of the focus added to the vertex’s distance h.

In this situation, the ordinate (or the y coordinate) of the parabola is 0 (zero) which means that there is no movement along the y axis.

Now let’s proceed in finding for the differential equation of the said parabola and this means that we need to eliminate the constant. In this situation, there are two constant (the p and h), and this means that the differential equation will be in the second derivative. So, the highest order derivative in the differential equation must be y’’(the second derivative of y).

SOLUTION:

First, use the “product rule” of the implicit equation which states that the derivative of a product of the two functions f(x) and g(x) is equal to the sum of the product of the first term and the derivative of the second term; and the product of the second term and the derivative of the first term. Or in representation of;

Get the first derivative,

Now, we get the equation which gives an answer depending on a constant p. But, remember that in finding differential equations, there shouldn’t be constant left. That’s why, let’s continue differentiating.
Use the product rule of implicit differentiation again, to eliminate the constant p.

Find the second derivative.

Since that the last equation doesn’t have any constant p and h anymore, therefore, the differential equation of the parabola which has a condition that the vertex and the focus must be lie on the x axis is;

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