What is the differential equation whose solution represents the family c(y + c)^2 = x^3?

(a) [2x/3 *(dy/dx) – y][x/3 *dy/dx]^2 = x^3

(b) [x/3 *(dy/dx) – y][x/3 *dy/dx]^2 = x^3

(c) [2x/3 *(dy/dx) – y][ 2x/3 *dy/dx]^2 = x^3

(d) [x/3 *(dy/dx) – y][ 2x/3 *dy/dx]^2 = x^3

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My question is taken from Linear First Order Differential Equations in portion Differential Equations of Mathematics – Class 12

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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