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

$f_y = \cos(x) \cos(y) = \cos(x) \cos(y) + \dfrac{dg}{dy}$

$\implies \dfrac{dg}{dy} = 0 \implies g(y) = C$

Then the general solution to the exact equation is

$f(x,y) = \boxed{-\ln|\cos(x)| + \cos(x) \sin(y) = C}$