Question

(10 points) Use the "mixed partials" check to see if the following differential equation is exact. If it is exact find a function F(x,y)F(x,y) whose level curves are solutions to the differential equation (−2exsin(y)−3y)dx+(−5x+excos(y))dy=0

Answer 1

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.