Find all solutions of the differential equation
![\[ y' + y \cot x = 2 \cos x \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-510312ddfe1677677e1911828c16d824_l3.png "Rendered by QuickLaTeX.com")
on the interval 
. Prove that exactly one of these solutions is valid on the larger interval 
.
Proof. We apply Theorem 8.3 (page 310 of Apostol) with
![\[ P(x) = \cot x, \qquad Q(x) = 2 \cos x. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-f9c09e2dad125725816d4a23a3ca6225_l3.png "Rendered by QuickLaTeX.com")
Further, we choose 
and obtain solutions in terms of 
, this gives us
![\[ A(x) = \int_{\frac{\pi}{2}}^x \cot t \, dt = \log (\sin x). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-8cccde13dfcd780f49ff175b991f5501_l3.png "Rendered by QuickLaTeX.com")
Therefore,
![\begin{align*} f(x) &= be^{- \log (\sin x)} + e^{-\log(\sin x)} \int_{\frac{\pi}{2}}^x 2(\cos t)e^{\log(\sin t)} \, dt \\[9pt] &= \frac{b}{\sin x} + \frac{1}{\sin x} \int_{\frac{\pi}{2}}^x \sin (2t) \, dt \\[9pt] &= \frac{b}{\sin x} - \frac{\cos (2x)}{2 \sin x} + \frac{1}{2 \sin x} \\[9pt] &= \frac{b}{\sin x} - \frac{1-2\sin^2 x}{2 \sin x} + \frac{1}{2 \sin x} \\[9pt] &= \frac{b}{\sin x} + \sin x. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-0c915c3bc33676f35d65719e6039f287_l3.png "Rendered by QuickLaTeX.com")
These are then all of the solutions valid on . The only one of these solutions valid on the interval is the one with 
, or 
I got C=b-1. The question doesn’t make sense ,however, since cot(x) isn’t defined everywhere. It would have made sense if the differential equation was sin (x)y’ + cos(x)y = sin(2x) instead.
Yeah, and regarding the holes in the solution I found that if you create a new function which fills in those holes to make the solution continuous everywhere, it is differentiable on R and satisfies the equation you’ve written here.
I found the following: f(x) = sinx + C/sinx, where C=b-1
I got C=b-1 like the others, but I want to say the the problem doesn’t make sense because cot(x) is not defined everywhere. it would have worked if the differential equation was sin(x)y’ + cos(x)y = sin(2x) instead.
I think the function is
or
where
The answer is slightly off: should be (b+1)/sinx + sinx , or C/sinx + sinx. cos(2*pi/2) is negative. So b=-1 ?
I’ve also found the following:
Same here