Find all solutions of the differential equation

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

Further, we choose and obtain solutions in terms of , this gives us

Therefore,

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