# DIIFERENTIAL EQUATION: $(x^2 + y^2) dx + x (3x^2 - 5y^2) dy = 0$

( x^2 + y^2 ) dx + x (3x^2 - 5y^2 ) dy = 0

### $(x^2 + y^2)\,dx + x(3x^2 -$(x^2 + y^2)\,dx + x(3x^2 - 5y^2)\,dy = 0$The variables are not separable The equation is not homogeneous Try:$\dfrac{dy}{dx} + \dfrac{x^2 + y^2}{x(3x^2 - 5y^2)} = 0\dfrac{dx}{dy} + \dfrac{x(3x^2 - 5y^2)}{x^2 + y^2} = 0$The equation is not linear. Try:$M = x^2 + y^2$→$\dfrac{\partial M}{\partial y} = 2yN = 3x^3 - 5xy^2$→$\dfrac{\partial N}{\partial x} = 9x^2 - 5y^2$The equation is not exact Try:$\dfrac{\dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x}}{N} = \dfrac{2y - (9x^2 - 5y^2)}{3x^3 - 5xy^2}\dfrac{\dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x}}{N} = \dfrac{2y - 9x^2 + 5y^2}{3x^3 - 5xy^2}$The equation does not have an integrating factor that is a function of x alone Try:$\dfrac{\dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x}}{M} = \dfrac{2y - (9x^2 - 5y^2)}{x^2 + y^2}\dfrac{\dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x}}{N} = \dfrac{2y - 9x^2 + 5y^2}{x^2 + y^2}\$

The equation does not have an integrating factor that is a function of y alone

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