Question

Find the differential equation of all the parabolas whose axes are parallel to the x-axis and latus rectum equal to a.

Answer 1

The differential equation of all parabolas whose axis are parallel to the x-axis and have latus rectum a is

$(y-\beta {)}^2=a(x-\alpha )$

Let two constant $\alpha$ & $\beta$

Thus we have to differentiate it twice

Differentiate both sides

$2(y-\beta )dy=a$ ..........$\left(1\right)$

Differentiating equation $\left(1\right)$ w.r.t. to x.

We get

$2(y-\beta )\frac{d^{2}y}{d{x}^2}+2{\left(\frac{dy}{dx}\right)}^2=0$ ...........$\left(2\right)$

Eliminating $\beta$ from equation $\left(1\right)$ and $\left(2\right)$, we have

$a\frac{d^{2}y}{d{x}^2}+2{\left(\frac{dy}{dx}\right)}^3=0$

Which is the required differential equation.