Your solution looks fine, I don't see how they could have done that either.
On a side note, you could also have used the Integrating Factor method with (and it would probably have been easier, since you already had it).
such that
The integrating factor is , so:
The point gives:
For the equation is separable.
Problem: The solution in the book gives for and I don't see how to find C_4 because the point isn't on the interval
Yeah, the family of functions is a solution for any value of C, and basically they have which is a constant. The problem is that you can't get to this value...
It turns out that this constant makes the piecewise defined solution continous at x=1 if you change the original intervals to and . The you can say:
However, this isn't correct because on the original interval. But I'm pretty sure I see their mistake now.