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:

.... (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).