Solving First Order Linear Differential Equation

[the0]

Homework Statement

Given (E): (x+1)^{2}(xy'-y) = -(2x+1)

Determine the set of applications from the interval I to ℝ which are solutions of (E) for:

a) I = (0,+∞)
b) I = (-1,0)
c) I = (-∞,-1)
d) I = (-1,+∞)
e) I = ℝ


The attempt at a solution

I have rearranged (E) into the form:

y'-\frac{y}{x} = \frac{-(2x+1)}{x(x+1)^{2}}

and chosen integrating factor:

exp(∫\frac{-1}{x}dx) = \frac{1}{x}

now multiplying by the integrating factor and integrating gives:

∫\frac{y'}{x}-\frac{y}{x^{2}}dx = ∫\frac{-(2x+1)}{x^{2}(x+1)^{2}}dx

partial fractions now give:

∫\frac{y'}{x}-\frac{y}{x^{2}}dx = ∫\frac{-1}{x^{2}}+\frac{1}{(x+1)^{2}}dx

After integrating and a tiny bit of algebra I get a general solution, for k\inℝ as:

y=\frac{1}{x+1}+kx

Now I think I am right in saying that this solution works fine for the intervals in a), b) and c)?

However for I = (-1,+∞) there is (or potentially is) a problem with the point 0 and for I = ℝ there are (or potentially are) problems with the points 0 and -1?

Now I'm not sure how to proceed. Am I thinking along the right lines?

Any help/pointers would be very much appreciated, Thanks!

 

[the0]

Can the general solution already posted simply extend to I = (-1,+∞) ??
If so how can this be shown in detail?

Also I don't yet understand what happens when we are considering ℝ

Thanks for any help!