As Ackbeet pointed out, unless there is some other condition we weren't told about, we are NOT justified in just dropping the absolute value. But we can say that

**if** x> 0, then |x|> 0 and the given solution is correct. But we could also say that if x< 0, |x|= -x and then integrating factor is x^4.

The critical point is that we have to choose one or the other- the solution cannot be continued past x= 0. Which solution is correct would depend on some additional fact- like an initial value if

, for some

, x will always be positive and and if

, x will always be negative.