Differential Equation: $(1-xy)^{-2} dx + \left[ y^2 + x^2 (1-xy)^{-2} \right] dy = 0$

Submitted by The Organist on

Hello, can anyone solve this equation?

I can't figure it out,

$(1-xy)^{-2} dx + \left[ y^2 + x^2 (1-xy)^{-2} \right] dy = 0$

Thanks.

Tags
Calculus
Differential Equation

Jhun Vert Wed, 12/23/2020 - 00:36

This is an exact equation
$(1 - xy)^{-2} \, dx + \left[ y^2 + x^2 (1 - xy)^{-2} \right] \, dy = 0$
 

Check for exactness:

$M = (1 - xy)^{-2}$

$\dfrac{\partial M}{\partial y} = -2(1 - xy)^{-3}(-x)$

$\dfrac{\partial M}{\partial y} = 2x(1 - xy)^{-3}$
 

$N = y^2 + x^2 (1 - xy)^{-2}$

$\dfrac{\partial N}{\partial x} = -2x^2(1 - xy)^{-3}(-y) + 2x(1 - xy)^{-2}$

$\dfrac{\partial N}{\partial x} = 2x^2y(1 - xy)^{-3} + 2x(1 - xy)^{-2}$

$\dfrac{\partial N}{\partial x} = 2x(1 - xy)^{-3} \left[ xy + (1 - xy) \right]$

$\dfrac{\partial N}{\partial x} = 2x(1 - xy)^{-3}$
 

$\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$, hence, exact!

 

To solve this type of equation, see this page: https://mathalino.com/node/494