A seductive equation sending her invitation in an "envelop"

Question

Solve the differential equation

x ( y' )³ − y( y' ) ² +1 = 0

Answer 1

it is Clairuat's 1st order non-linear

y = x y' + 1/(y')^2 = xy' + f( y') where f(t) = 1/t^2

the general solution is y = cx + f(c)

the singular envelopes are x = -f'(c) and y = f(c) - c f'(c)

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