If you put this in standard from and get the integrating factor you will find that it is (x+1). It will turn back into what you started with we notice that the left hand side is the the implicit derivative

So we get

so now we integrate both sides to get

using the intial condition we get

So we get

For the interval It depends on if you had

If it is the first then solution only exits for

If it is the 2nd the only singular point is x=-1 so the two intervals would be

So the largest would be