Use the "mixed partials" check to see if the following differential equation is exact. if it is exact find a function f(x, y) whose level curves are solutions to the differential equation. (−3xy^2+y)dx + (−3x^2y+x)dy =0

We have

and

so the equation is indeed exact. So we want to find a function such that

Integrating both sides of the first equation wrt gives

Differentiating both sides wrt gives

So we have

or

step-by-step explanation:

in the absence of parentheses, i will have to make some assumptions:   i interpret your (3x+5/y^6)(4y^3/5)​ to mean:

3x + 5     4y^(3/5)       (3x + 5)(4)

* =

    y^6             1               y^(6-3/5)

if this is not what you expected, repost your question using parentheses to indicate whether you meant

4y^3

    or   4y^(3/5)

    5