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

Answer:

The differential equation is not exact.

Step-by-step 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

$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$

But

$\frac{\partial M}{\partial y}=\frac{\partial (3x^3+2y-2x-y^2)}{\partial y}=2-2y$

whereas

$\frac{\partial N}{\partial x}=\frac{(-1)}{\partial x}=0$

and we can see

$\frac{\partial M}{\partial y}\neq \frac{\partial N}{\partial x}$

So the differential equation is not exact.