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Math Help - Second order DE

  1. #1
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    Second order DE

    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..
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  2. #2
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    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?
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  3. #3
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    The integarting factor I found was e^ integral of (1/t -1) which i worked out to be te^-t but then when i multiplied it with t/(t+1) I had no idea how to integrate it....
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  4. #4
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    Right. Looking good so far. So now we must integrate the RHS which is, as you've pointed out, the integral

    \displaystyle{\int\frac{t^{2}e^{-t}}{t+1}\,dt.}

    This is not a very nice integral, I grant you. At this point, I would probably substitute s=t+1, in order to get rid of that sum in the denominator. What does this get you?
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  5. #5
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    s = t + 1
    t = s -1
    dt = ds
    [(s-1)^2e^-(s-1)]/s ?
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  6. #6
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    That's right. Keep going... what would you do next?
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  7. #7
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    uuuummm I tried to split it up as in (s-1)^2/s x e^1-s/2 then expanded the (s-1)^2/2 to get (s - 2 - 1/s). e^(1-s)/s ...but that didnt really help haha
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  8. #8
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    Hmm. Clean up your notation there and check your minus signs. The integral should break up into three pieces, each of which you handle differently.
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  9. #9
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    Okay I redid it and got exp(1-s) - 2exp(1-s).s^-1 + s^2exp(1-s) I can only integrate 2 of them though..I'm not quite sure how to integrate the middle term..
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  10. #10
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    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.
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