Question

Determine whether the given differential equation is exact. If it is exact, solve it. (tan(x)-sin(x)sin*y))dx+cos(x)cos(y)dy=0 g

Answer 1

The differential equation

`M(x,y) \, dx + N(x,y) \, dy = 0`

is considered exact if
`M_y = N_x` (where subscripts denote partial derivatives). If it is exact, then its general solution is an implicit function
`f(x,y)=C` such that
`f_x=M` and
`f_y=N`.

We have

`M = \tan(x) - \sin(x) \sin(y) \implies M_y = -\sin(x) \cos(y)`

`N = \cos(x) \cos(y) \implies N_x = -\sin(x) \cos(y)`

and
`M_y=N_x`, so the equation is indeed exact.

Now, the solution
`f` satisfies

`f_x = \tan(x) - \sin(x) \sin(y)`

Integrating with respect to
`x`, we get

`\displaystyle \int f_x \, dx = \int (\tan(x) - \sin(x) \sin(y)) \, dx`

`\implies f(x,y) = -\ln|\cos(x)| + \cos(x) \sin(y) + g(y)`

and differentiating with respect to
`y`, we get

`f_y = \cos(x) \cos(y) = \cos(x) \cos(y) + (dg)/(dy)`

`\implies (dg)/(dy) = 0 \implies g(y) = C`

Then the general solution to the exact equation is

`f(x,y) = \boxed\cos(x)`