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**HallsofIvy** 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 $\displaystyle x(t_0)> 0$, for some $\displaystyle t_0$, x will always be positive and and if $\displaystyle x(t_0)< 0$, x will always be negative.