A challenging ODE question: find an integrating factor

[AdrianZ]

This is the ODE: y' + siny + xcosy + x = 0.
The problem is: Find an integrating factor for the ODE above.

You can see my solution to the ODE here: https://www.physicsforums.com/showthread.php?t=543662. from my solution it seems that $$e^x(sec^2(y/2))$$ must be an integrating factor. but I fail to show that it's an integrating factor.
I guess it must be truly challenging to find an integrating factor for it :(

 

[lurflurf]

I think that is an integrating factor, recall the general test
u is an integrating factor of N y'+M if

M_y-N_x=(u_x/u)N-(u_y/u)M
where subscripts are partial derivatives and for this equation we have
M=sin(y) + x cos(y) + x
N=1
u=e^xsec²(y/2)=2e^x/(1+cos(y))
you will need to use some simple trigonometric identities

 

[AdrianZ]

I think that is an integrating factor, recall the general test
u is an integrating factor of N y'+M if

M_y-N_x=(u_x/u)N-(u_y/u)M
where subscripts are partial derivatives and for this equation we have
M=sin(y) + x cos(y) + x
N=1
u=e^xsec²(y/2)
you will need to use some trigonometric identities for half angles

What he says doesn't make sense to me to be honest. I've solved the ODE, and I'm 99% sure that e^xsec²(y/2) is an integrating factor. he tells me that I should multiply the integrating factor and show that it turns the ODE into an exact differential because his problem asks me to find an integrating factor for it, not solving the ODE.

 

[lurflurf]

Yes if M_y-N_x=(u_x/u)N-(u_y/u)M
then the differential is exact this follows from requiring
(uM)_y-(uN)_x=0
then it is possible to choose a P such that
N=P_y/u
M=P_x/u
hence
(P)'/u=N y'+M
as desired

 

[blue_raver22]

I understand the difficulty of finding an integrating factor for a given ODE. It requires a deep understanding of mathematical concepts and techniques. In this particular case, the ODE is non-linear and contains both trigonometric and exponential terms, making it even more challenging to find an integrating factor.

After reviewing your solution and the discussion on the forum, it seems that you have correctly identified e^x(sec^2(y/2)) as a potential integrating factor. However, you are struggling to prove that it is indeed an integrating factor. I recommend taking a closer look at the properties of integrating factors and how they affect the given ODE. This may help you in your proof.

Additionally, I would suggest consulting with other experts or seeking guidance from a mathematics tutor to further explore this problem. Sometimes, collaborating with others can lead to new insights and approaches.

Overall, finding an integrating factor for a challenging ODE like this requires persistence, patience, and a deep understanding of mathematical concepts. Keep exploring and don't give up, as the solution to this problem may lead to new discoveries and insights in your field of study.