Find the general solution of the differential equation (regarding x as depending

Question

Find the general solution of the differential equation
(regarding x as depending variable and y as independent). Primes denote
derivatives with respect to x:

y'=2(xy'+y)y^(3)

Need the step by step breakdown please detailed

Explanation / Answer

f_x = 8x - y - 3x^2, f_y = 2y - x, Setting these equal to 0: y = 8x - 3x^2 and y = x/2. ==> 8x - 3x^2 = x/2. ==> x= 0 or 5/2. Thus, we have two critical points: (x,y) = (0,0) and (5/2, 5/4). To classify these, we take second order derivatives: f_xx = 8 - 6x, f_yy = 2, f_xy = -1. D = (f_xx)(f_yy) - (f_xy)^2 = 15 - 12x. At (0,0), D > 0 and f_xx > 0. So, this is a local minimum. At (5/2, 5/4), D < 0. So, this is a saddle point.

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