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

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

Tags
Differential Equation, DE

Jhun Vert Sun, 07/17/2016 - 12:57

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

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:

$M = x^2 + y^2$   →   $\dfrac{\partial M}{\partial y} = 2y$

$N = 3x^3 - 5xy^2$   →   $\dfrac{\partial N}{\partial x} = 9x^2 - 5y^2$

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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