DE Exactness: (2x+y)dx-(x+6y)dy=0?

[Eastonc2]

Homework Statement

determine whether the DE, (2x+y)dx-(x+6y)dy=0, is exact

Homework Equations

i understand how to determine if they are exact, I just don't understand this specific instance. for my case, M(x,y)=2x+y, but would N(x,y)=(x+6y), or (-x-6y)?

The Attempt at a Solution

using my first N(x,y), the equation is exact, however, using the second, they are not exact.

Just need clarification at this point

 

[I like Serena]

Hi Eastonc2!

Your N(x,y)=(-x-6y).
That is how it matches the definition of an exact DE.

But how did you determine that it was exact with the first N(x,y)?
Because I don't think it is.

 

[Eastonc2]

ah, my fault, M(x,y)=2x+y

i was looking at the equation above it for that first part. the second part is correct though.

 

[I like Serena]

Ah, now I see your dilemma.

To make sure, perhaps you should try to find a function of which the partial derivatives match with M(x,y) and N(x,y).
Can you find such a function?

 

[LCKurtz]

Have you studied the case where M and N are homogeneous of the same degree, as these are?