Find the differential equation for a parabola with latus rectum 4a and whose axis are parallel to x-axis.
The equation of the parabola satisfying the given requirements is (y-k)^2 = 4ax. So the solution of the DE is
(y - k)^2 = 4ax .... (1)
=> y^2 - 2ky + k^2 = 4ax .... (2)
Differentiate (2) with respect to x:
$\displaystyle 2y \, y' - 2k\, y' = 4a$ .... (3)
Obviously k needs to be eliminated from (3). k can be got from (1) in terms of x, y and a. Substitute this result into (3).