Question

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 differential, dF(x,y) gives the differential equation. That is, level curves F(x,y)=C are solutions to the differential equation: dydx=3x3+2y−2x−y2 First rewrite as M(x,y)dx+N(x,y)dy=0 where M(x,y)= equation editor Equation Editor , and N(x,y)= equation editor Equation Editor . If the equation is not exact, enter not exact, otherwise enter in F(x,y) as the solution of the differential equation here

Answer 1

The differential equation is not exact.

Explanation:

We can write the differential equation as

`(3x^3+2y-2x-y^2)dx+(-1)dy=0`

with

`M(x,y)= 3x^3+2y-2x-y^2`

`N(x,y)=-1`

to check whether the differential equation is exact, we must verify that

`(\partial M)/(\partial y)=(\partial N)/(\partial x)`

But

`(\partial M)/(\partial y)=(\partial (3x^3+2y-2x-y^2))/(\partial y)=2-2y`

whereas

`(\partial N)/(\partial x)=((-1))/(\partial x)=0`

and we can see

`(\partial M)/(\partial y)\\eq (\partial N)/(\partial x)`

So the differential equation is not exact.