Question

Differential equation of all parabolas having their axis of symmetry coinciding with the x axis is

Answer 1

Let P denote the family of parabolas and let (a, 0) be the

focus of a member of the given family, where a is an arbitrary constant. Therefore, equation

of family P is

y^2 =4ax ... (1)

Differentiating both sides of equation (1) with respect to x, we get

2y.dy/dx =4a ... (2)

Substituting the value of 4a from equation (2)

in equation (1), we get

y^2 = 2y( dy/dx)(x)

or

y^2 2xy dy/xy dx  =0

which is the differential equation of the given family

of parabolas.