Differential Equations

Question

Determine whether the given differential equation is exact. If it is exact, solve it. (If it is not exact, enter NOT.)

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

Answer 1

Yes, it is.

After dividing everything by dx,

M(x,y) = tanx - sinx sin y and N(x,y) = cos x cos y

dM/dy = -sinx cos y while dN/dx = -sin x cosy y

S(x,y) = integral ( M(x,y) dx = integral ( tan x) dx - integral ( sin x sin y) dx + h(y) where

h(y) is the constant function of integration in terms of y

the anti-derivative of tan x is - ln | cos x|

and the anti-derivative of sin x = - cos x

S(x,y) = - ln | cos x| + cos x sin y + h(y)

dS/dy = -cos x cos y + h'(y) = cos x cos y = N(x,y)

so h'(y) = 0 ---> h(y) = k is a constant

The general solution is S(x,y) = - ln | cos x| + cos x cos y + k