Differential equation of all parabolas having their axes of symmetry coincident with the axis of X is :

Options

A. `yy_(1)^(2)+y_(2)=0`
B. `yy_(2)+y_(1)^(2)=0`
C. `y_(1)^(2)+yy_(2)=0`
D. `yy_(2)+y_(1)=0`

Answer

B

Explanation

The equation that represents a family of parabolas having their axis of symmetry coincident with the axis of x is
`y^(2)=4a(x-h)" …(i)"`
This equation contains two arbitrary constants, so we shell differentiate it twice to obtain a second order differential equaiton
Differentiating (i) w.r.t. x, we get
`2y(dy)/(dx)=4arArr y (dy)/(dx)=2a" ...(ii)"`

Differentiating (ii) w.r.t. x, we get
`y(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)=0rArryy_(2)+y_(1)^(2)=0`.
which is the required differential equation.