Question

What is the differential equation of a family of parabolas with the foci at the origin and axis along the X-axis?

(a) 2xy’+ 4y(y’)^2-y=0

(b) xy’+ y(y’)^2-y=0

(c) 2xy’+ y(y’)^2-y=0

(d) 2xy’+2y(y’)^2-y=0

I have been asked this question during a job interview.

My query is from Formation of Ordinary Differential Equations by Elimination of Arbitrary Constants in portion Ordinary Differential Equations – First Order & First Degree of Engineering Mathematics

Answer 1

The correct choice is (c) 2xy’+ y(y’)^2-y=0

Easiest explanation: The equation is, y^2=4ax+4a^2……………………………………. (1)

Differentiating (1) with respect to x, we get,

2yy’=4a ………………………………………………………………………………………….. (2)

Therefore, substituting the value of 4a in (1), we get,

y^2=2yy’x+(yy’)^2

So, the required differential equation is given by,

2xy’+y(y’)^2-y=0