# Thread: finding a value to make a DE exact

1. ## finding a value to make a DE exact

this is the question: Find the value for k so that the given DE is exact

(6x (y^3) + cos y )dx + (2k (x^2) (y^2) - x sin y)dy =0

thank you so much

2. Originally Posted by kithy

this is the question: Find the value for k so that the given DE is exact

(6x (y^3) + cos y )dx + (2k (x^2) (y^2) - x sin y)dy =0

thank you so much
For a DE to be exact, the following must be true:

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

From your DE, we see that:

$\displaystyle M(x,y)=6xy^3+\cos y\implies \frac{\partial M}{\partial y}=18xy^2-\sin y$

$\displaystyle N(x,y)=2kx^2y^2-x\sin y\implies \frac{\partial N}{\partial x}=4kxy^2-\sin y$

Thus,

$\displaystyle 18xy^2-\sin y=4kxy^2-\sin y$

Try taking it from here.

Hope that this makes sense!

--Chris

3. Ok, i equated the two equations and i came out with K= (18xy^2)/(4xy^2) and so i am assuming the xy^2 cancel out and we are left with k=9/2 is that right?

4. Originally Posted by kithy
Ok, i equated the two equations and i came out with K= (18xy^2)/(4xy^2) and so i am assuming the xy^2 cancel out and we are left with k=9/2 is that right?
Sorry for the late response. I've been in school all day...

Yes, that is correct. Does is make more sense now?

--Chris

5. Yes it really does, thank you so much, i understand it better now