Question

Find the differential equation of all the parabola having axis parallel to the $x-$axis.

Answer 1

Equation of parabola having axis parallel to x-axis is given by,

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

differentiating on both sides, we get,

$2(y-\beta )=\frac{dy}{dx}=a$

again differentiating on both sides, we get,

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

eliminating $\beta$ from the above equations, we get,

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

Is the required differential equation.