Well, you can reduce the order immediately by setting v = u', as you have already (sort of) mentioned. Then I would probably use the integrating factor on the resulting DE:
v' - v (t-1)/t = t/(t+1).
What is the integrating factor here?
Which method can i use to solve u'' - (t-1)/t u' = t/(t+1)??
This is an equation I have formed by making it U absent and I let p = dy/dx and dp/du = d^y/dx^2 but then i'm stuck after that -.-"" any help would be appreciated I've attempted this question so many times...I tried using an integrating factor to solve but i couldnt do it..
Right. Looking good so far. So now we must integrate the RHS which is, as you've pointed out, the integral
This is not a very nice integral, I grant you. At this point, I would probably substitute in order to get rid of that sum in the denominator. What does this get you?
Right. The middle term is the tricky one. In fact, it's so hard it's easy (if you know the result, that is).
The middle integral gives you what's called the exponential integral function. In fact, there is no elementary antiderivative. So your final solution is just going to have to include that term unsimplified. Incidentally, wolframalpha has the exponential integral term in its solution if you just go from scratch with the DSolve command.
Don't forget also that you have to integrate one more time to get to u. I wouldn't bother trying to integrate the exponential integral function. Just write out the integral.