Question

Verify that the given differential equation is an exact: 

(y²-1)dx+(2xy+3y)dy=0

Answer 1

$$(y^2-1)dx+(2xy+3y)dy=0$$

$$P(x, y)=y^2-1, \dfrac{\partial P}{\partial y}=2y$$

$$Q(x, y)=2xy+3y, \dfrac{\partial Q}{\partial x}=2y$$

 Use the test for exactness:

$$\dfrac{\partial Q}{\partial x}= \dfrac{\partial P}{\partial y}$$

Since

$$\dfrac{\partial Q}{\partial x}=2y= \dfrac{\partial P}{\partial y}$$

the differential equation

$$(y^2-1)dx+(2xy+3y)dy=0$$

is an exact equation.