Use the "mixed partials" check to see if the following differential equation is exact. If it is exact find a function (x,y) whose differential, (x,y) gives the differential equation. That is, level curves (x,y)= are solutions to the differential equation: yx=−3x4+2y−2x−4y2 First rewrite as (x,y)x+(x,y)y=0 where (x,y)= , and (x,y)= . If the equation is not exact, enter not exact, otherwise enter in (x,y) as the solution of the differential equation here =.

The differential equation is exact.

So the solution of the differential equation is

Step-by-step explanation:

So the question is to check whether or the given differential equation

is exact

Step One : Put the differential equation in standard form

To check whether it is exact or not we are going to put it in form by cross-multiplication

Since all the terms in the dx side are negative we are going to move them to the other side to get everything on the same side so we have

and this is in the standard form which is

Step Two: Take the partial differentiation of the standard form and compare it there are equal

Since their partial differential is equal then the differential equation is exact

Step Three : Solve the differential equation

Now that we know it is exact how do we solve it well we are going to take

of which is

The reason for the  C(y) is because the equation is been integrated with respect to x so any function in y will have derivative 0 with respect to x so we are going to be left with some arbitrary function in y

Now to figure this arbitrary function C(y) we are going to take the derivative of the function with respect to y and then compare it to what it is suppose to be which is N

and

So we have

=>    

=>    

Taking integral of both sides

put it back into equation one we have