Solve the exact differential equation (tan(x) − sin(x) sin(y)) dx + cos(x) cos(y) dy = 0.

Solution:

The differential equation is an equation involving derivatives of an unknown function. 

The differential equation M(x, y) dx + N(x, y) dy = 0 is exact when ∂M/∂y = ∂N/∂x.

Given that (tan(x) − sin(x) sin(y)) dx + cos(x) cos(y) dy = 0

Let M(x, y) = tan(x) − sin(x) sin(y) and N(x, y) = cos(x) cos(y).

∂M/∂y = - sin(x) cos(y) = ∂N/∂x

To find the solution, we solve ∂F/∂x = M(x, y) = tan(x) − sin(x) sin(y)

F(x, y) = -log|cos x| + cos(x) sin(y) + G(x, y)

∂F/∂y = N(x, y)

cos(x) cos(y) + G'(x, y) = cos(x) cos(y)

G'(x, y) = 0

G(x, y) = C, where C is an arbitrary constant.

Thus, the solution F(x,y) = -log |cosx| + sin (y) cos (x) = C

Solve the exact differential equation (tan(x) − sin(x) sin(y)) dx + cos(x) cos(y) dy = 0.

Summary:

The solution is the exact differential equation (tan(x) − sin(x) sin(y)) dx + cos(x) cos(y) dy = 0. is -log |cosx| + sin (y) cos (x) = C