First order non-linear non-homogeneous DE problem

[achacttn]

Hello. I am trying to solve this problem methodically but my solution does not seem to agree with the given answer.

Homework Statement

The differential equation is:
(sinx)y' - (cosx)y = 1 + C

Homework Equations

The Attempt at a Solution

y' - (cosx/sinx)y = 1/sinx + C/sinx

When finding the integrating factor, I used:
e^-∫(cosx/sinx)dx
I was wondering if this was either
|sinx|^(-1) or -|sinx| and why? I tried solving the equation using both methods

1st method (using |sinx|^(-1))

(|sinx|^(-1))y' - cosx/((sinx)^2).y = 1/(sinx)^2 + C/(sinx)^2
LHS becomes
D[|sinx|^(-1).y] = 1/(sinx)^2 + C/(sinx)^2
Integrating both sides and multiplying by |sinx| gives
y = x/sinx + Cx/sinx + Dsinx

When using -|sinx| I get:
y = x/sinx Cx/sinx + D/sinx

The given solution is
y = -x.cosx + sinxln(|sinx|) - Acosx + Bsinx

I have no idea how to get this. Any help would be appreciated, thanks!

Edit: I see my error for using the |sinx|^(-1), I didn't integrate the right hand side properly.

I get |sinx|^(-1).y = -cotx + Ccotx + D
so
y = -xcosx + Ccosx + Dsinx, which is still not the right answer

 

[dikmikkel]

It is in fact linear and on the form:
$y' + f(x)y+g(x) = 0$
The solution should be as follows:
$y(x) = \left(\int g(x)e^{\int f(x)\text{d}x}\text{d}x+K_1\right)e^{\int (-f(x))\text{d}x}$