Correct choice is (c) [2x/3 *(dy/dx) – y][ 2x/3 *dy/dx]^2 = x^3
The best I can explain: The given family is c(y + c)^2 = x^3
Differentiating once, we get
c[2(y + c)]dy/dx = 3x^2
=> 2x^3 (y + c)/(y + c)^2 * dy/dx = 3x^2
=> 2x^3/(y + c) * dy/dx = 3x^2
Or, 2x (y + c)/(y + c)^2 * dy/dx = 3
=> 2x/3 *[dy/dx] = (y + c)
=> c = 2x/3 *[dy/dx] – y
Substituting c back into equation (1), we get
[2x/3 *(dy/dx) – y][ 2x/3 *dy/dx]^2 = x^3
which is the required differential equation