Answer:
The equation is not exact.
Step-by-step explanation:
Given the differential equation:
(-2e^x sin(y) - 3y)dx - (-5x + e^x cos(y))dy = 0
We need this equation to be in the form:
Mdx + Ndy = 0
Where M = M(x, y) and N = N(x, y)
to verify its exactness.
So, rewriting the equation, we have
(-2e^x sin(y) - 3y)dx + (5x - e^x cos(y))dy = 0
Now, if ∂N/∂x = ∂M/∂y, the equation is exact.
Here, M(x, y) = (-2e^x sin(y) - 3y), and N(x, y) = (5x - e^x cos(y))
(∂/∂x)(-2e^x sin(y) - 3y) = -2e^x sin(y)
(∂/∂y)(5x - e^x cos(y)) = e^x sin(y)
Since ∂N/∂x ≠ ∂M/∂y, we say the equation is not exact, and hence, we are not required to find its general solution.
