Question

x^2dy/dx-3y^4=2xy

Answer 1

$$x^2\dfrac{dy}{dx}-3y^4=2xy$$

Let $u=y^3.$ Then $u'=3y^2y'$

$$3x^2y^2y'-9y^6=6xy^3$$

$$x^2u'-6xu=9u^2$$

$$u'-\dfrac{6}{x}u=\dfrac{9}{x^2}u^2$$

Bernoulli Equation

$$z=u^{1-2}=u^{-1}$$

$$z'=-u^{-2}u'$$

$$z'+\dfrac{6}{x}z=-\dfrac{9}{x^2}$$

$$x^6z'+6x^5z=-9x^4$$

$$d(x^6z)=-9x^4dx$$

$$x^6z=-\dfrac{9x^5}{5}+\dfrac{C}{5}$$

$$z=\dfrac{C-9x^5}{5x^6}$$

$$y^3=\dfrac{5x^6}{C-9x^5}$$