Question

What am I doing wrong?
Determine whether the given differential equation is exact. If it is exact, solve it. (If it is not exact, enter NOT.)
(tanx − sinxsiny)dx + cosxcosy dy = 0
My work:
pM/py= tan x-sinxsiny=-sinxcosy
pM/py=cosxcosy=-cosysinx
Therefore this is exact.
Solve.
f(x,y)=integral of (tanx-sinxsiny)dx+g(y)
f(x,y)=cosxsiny-ln(cosx)+c_1+g(y)
cosxsiny-ln(cosx)+c_1=c_2
Final Answer: cos(x)sin(y)-ln(cos(x))=C
But my answer is wrong and I don't know why.

Answer 1

looking at this form a physical sciences perspective, the big problem i think you have is that the field is full of singularities due to the fact it has a tan(x) in there.

so even though you get the mixed partials to be equal, and you solve for a potential function, the potential function is not a potential function.

if that means anything to you, read on here

http://mathinsight.org/conservative_vector_field_determine

Answer 2

actually, a more mathsy answer might look at the fact that the underlying potential function is expressed as \(\Phi (x,y,z) = const\). it is from this fact that we can arrive at exact differential equations, because it follows that
\[d\Phi = \Phi_x dx + \Phi_y dy + \Phi_z dz = 0\]

and that's where you get the test for the mixed partials and exactness

however, the singularities means you cannot claim that \(\Phi (x,y,z) = const\) in the first place

in case i am leading you astray, let me tag a few people @ganeshie8 @freckles @Loser66

Answer 3

on the other hand, wolfram likes your answer
http://www.wolframalpha.com/input/?i=%28tanx+%E2%88%92+sinxsiny%29dx+%2B+cosxcosy+dy+%3D+0

Answer 4

OK, first of all, my bad in my first post: the tans are in the DE and not the potential, but the point still holds. in the first partial they are still signalling singularities in the underlying potential function so the claim \(\Phi = const\) cannot be made without restrictions on the domain being made also.

so, for example, what happens when , say, \(x = \pi/2\) in @nelly_416 's \(cos(x)sin(y)-ln(cos(x))=C\) solution

it is also interesting that the MW solution is written in a very particular way but can be reorganised to be @nelly_416 's solution.

you can buy your way out of these problems by re defining the values at erstwhile singularities. so is that the problem?

Answer 5

Thank you! The final answer I received was ln|secx|+cosxsiny
Which turned out to be correct.

Answer 6

lol!!!

so \(-ln(cosx) = ln|secx|\) was the problem?!?!

still think this is messed up because x is allowed to be \(\pi / 2\) many times over. but great that closure has been reached, & well done for persevering!

:p

Answer 7

out of interest, were you typing these answers into a computer?

Answer 8

Yes, the homework was online.