Question

The differential equation of all parabolas each of which has a latus rectum 4a and whose axes are parallel to the x-axis is of order 2 degree_________

Answer 1

Answer: 1

Equation to the family of parabolas is $(y-k{)}^2=4a(x-h)$
$2(y-k)\frac{dy}{dx}=4a$ (differentiating w.r.t. $x$)
or $(y-k)\frac{dy}{dx}=2a...(1)$
or $\left(y-k\right)\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2=0$ (differentiating w.r.t. $x$)
or $2a\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^3=0$ [substituting $y-k$ from equation (1)]
Hence, the order is $2$ and the degree is $1$