Form a differiantial equation by eliminating the constant A in the equation y = A (x-A) ^ 2.?

Question

i tried it by finding dy/dx [dy/dx = 2A(x-A)] then comparision with above equation by finding (dy/dx)^2/y got the value of A and substitued the value of A but the answer does not tally. If any of you are solving by using determinants please explain. also let me know why my method is not giving me the solution

I am looking at a solution without d^2/dy2
x,y,dy/dx, (dy/dx)^2 are admissible. mathematically speaking one constant A menas we can have a solution without going to the second derivative level.

Answer 1

y= A (x-A)^2
differentiate
dy/dx = A 2(x-A)
square both sides

(dy/dx)^2 = 4A^2(x-A)^2
= 4Ay...1


differentiate onece again

2(dy/dx)d^2/dx^2 = 4A dy/dx

or 2(dy/dx)(d^2/dx^2) = 4A = 4(dy/dx)^2/y

or 2(d^2/dx^2) = 4(dy/dx)/y

Answer 2

First of all, there are of course different diff. eq. possible that will have the same function as solution.
The easiest way I see is to differentiate twice, which gives
y' = 2A (x-A) and y'' = 2 A. Substitute these in y=A(x-A)^2 to get
4 y = y' (2x-y''), but of course other solutions are possible.