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

Sydney Sales's picture

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

Jhun Vert's picture

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

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

Wala pa akong nakitang solution. Kung meron ka na, pease share.

Sydney Sales's picture

eto po yung answer:

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

Jhun Vert's picture

Are you sure your equation is correct? And based on your answer key, there should be an initial condition because there is no constant c in your answer.

The equation is wrong it should be y(x^2+y^2)dx+x(3x^2-5y^2)dy=0, when x=2 , y=1

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