method of solution of differential equation in general form first Solve in detail after proving on the equation.

Question

(𝑦 ′ )^3 . 𝑥 − (𝑦 ′ )^2 . 𝑦 − 1 = 0

Answer 1

We solve this equation for y = xy' - 1/(y')².

This is Clairaut equation.

General solution of this equation y = Cx - 1/C², where C is arbitrary constant

Singular solution in parametric form: x = - (-1/p²)' = - 2/p³

y = xp - 1/p²