The differential equation of the family of straight lines whose slope is equal to y - intercept ,is

Options

A. `a(d^(2)y)/(dx^(2))+((dy)/(dx))^(3)=0`
B. ` 2a (d^(2)y)/(dx^(2))+((dy)/(dx))^(3)=0`
C. ` 2a (d^(2)y)/(dx^(2))- ((dy)/(dx))^(3)`
D. ` a (d^(2)y)/(dx^(2))-((dy)/(dx))^(3)=0`

Answer

b

Explanation

Equation of the family of such parabola is
` ( y - k)^(2) = 4a (x-h) " "` …(i)
where h and k are arbitrary constants ltbRgt On differetiating w.r.t , we get
`(y-k) (dy)/(dx) = 2a " " ` (ii)
Again , on differentiating , we get
` (y - k) (d^(2)y)/(dx^(2)) + ((dy)/(dx))^(2)=0" "`…(iii)
On putting hte value of `(y-k) ` from Eq. (ii) in Eq. (iii) , we get
` 2a (d^(2)y)/(dx^(2)) + ((dy)/(x))^(3) = 0 ,`
which is the required differential equation