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The differential equation that represents all parabolas having their axis of symmetry coincident with the axis of x, is

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`

Text Solution

Verified by Experts

The correct Answer is:
B
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.
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