# Determine whether the given differential equation is exact. (\tan x - \sin x \sin y)dx + \cos x...

## Question:

Determine whether the given differential equation is exact.

{eq}\displaystyle (\tan x - \sin x \:\sin y)\: dx + \cos x \cos y \: dy = 0 {/eq}

## Exact Differential Equation:

The differential of {eq}f(x,y) {/eq} is {eq}f_x(x,y)\, dx+f_y(x,y)\, dy {/eq}. So {eq}f(x,y)=k {/eq} would be a solution to the differential equation {eq}f_x(x,y)\, dx+f_y(x,y)\, dy=0 {/eq}.

We say that a differential equation {eq}M\, dx+N\, dy=0 {/eq} is exact if it is of the form {eq}f_x\, dx+f_y\, dy=0 {/eq}.

To test we note that {eq}M\, dx+N\, dy=0 {/eq} is exact if and only if {eq}M_y=N_x {/eq}.