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Differential equation and initial value

  1. Sep 23, 2009 #1
    1. The problem statement, all variables and given/known data

    [img=http://img3.imageshack.us/img3/3417/questionec.th.jpg]


    2. Relevant equations



    3. The attempt at a solution

    I tried dividing both sides by (x^5+1), however the integration becomes really complex... can someone give me suggestions on how to do this?
     
  2. jcsd
  3. Sep 23, 2009 #2

    rock.freak667

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    put the -10x4y on the left side then divide by tr x5+1. Integrating factor it.
     
  4. Sep 24, 2009 #3
    that's what I did and then I need to integrate -10x^4/(x^5+1) dx right and then then to get the integrating factor is just e to the power of whatever the result of the integration is... however the integration is quite hard...
     
  5. Sep 24, 2009 #4

    rock.freak667

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    alright now, so


    [tex]\int \frac{-10x^4}{x^5+1} dx[/tex]

    see how d/dx(x5+1)=5x4 ?

    Can you use a substitution to make this integral easier?
     
  6. Sep 24, 2009 #5
    Okay say I solve the integral and then the integrating factor would be e to the power of this resulting integral right? so then what do I need to do next in order to solve for this problem?
     
  7. Sep 24, 2009 #6

    rock.freak667

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    right so for y'+P(x)y=Q(x), when you multiply by an integrating factor 'u', the left side becomes d/dx(uy) . That's why we multiply by u in the first place.


    So you'll need to basically integrate uQ(x) w.r.t. x
     
  8. Sep 24, 2009 #7
    and then divide that by the integrating factor right?
     
  9. Sep 24, 2009 #8

    rock.freak667

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    yes you can if you feel the need to.
     
  10. Sep 24, 2009 #9
    what do you mean by if I need to? doesn't it always works like that?
     
  11. Sep 24, 2009 #10
    this now comes to:

    [tex]\int \frac{x^2+5x-4}{x^5+1} e^{-2ln(x^5+1)} dx[/tex]

    I guess this can be simplified to:

    [tex]\int \frac{x^2+5x-4}{(x^5+1)^3} dx[/tex]

    is this true?

    how can I solve this such complex integration?
     
    Last edited: Sep 25, 2009
  12. Sep 25, 2009 #11
    anyone please?
     
  13. Sep 25, 2009 #12

    rock.freak667

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    Re do your integrating factor as it was to integrate 10x4/(x5+1) not with the -ve sign.
     
  14. Sep 25, 2009 #13
    Integrating [tex]\frac{10x^4}{(x^5+1)}[/tex] the result I got is [tex]2ln(x^5+1)[/tex] and so the integrating factor is [tex]e^{2ln(x^5+1)}[/tex] which simplifies to [tex](x^5+1)^2[/tex].

    Then I do [tex] \int (x^2+5x-4)(x^5+1) [/tex] and the result of this integration I divide by [tex](x^5+1)^2[/tex] which is the integrating factor. Is this the correct step to find the solution?

    Please correct me if I am wrong.
     
    Last edited: Sep 26, 2009
  15. Sep 25, 2009 #14

    rock.freak667

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    yes just integrate and divide, also don't forget the constant of integration
     
  16. Sep 26, 2009 #15
    by the constant of integration you mean the C right? The terms after integration that I found is very long.....
     
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