Question

Q\Determine the following ODEs are exact or not? Why?
(y+cosx)dx+(x+siny)dy=0

Answer 1

$\displaystyle (y+\cos{x})\mathrm{d}x+(x+\sin{y})\mathrm{d}y=0\\ M = y + \cos{x}\\ \frac{\partial M}{\partial y} = 1\\ N = x + \sin{y}\\ \frac{\partial N}{\partial x} = 1\\ \textsf{Therefore, since}\, \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}, \\ \textsf{the differential equation is exact.}$