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Solve the difference equation?

  1. Jul 13, 2013 #1
    Solve the difference equation yn+1=(n+1)/(n+2) yn in terms of the initial value y0.
     
  2. jcsd
  3. Jul 13, 2013 #2

    tiny-tim

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    Hi Success! :wink:

    Show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:

    (common-sense should solve this)
     
  4. Jul 13, 2013 #3
    I really don't even know how to start. I guess you should begin with yn=1/(n+1) y0.
     
  5. Jul 13, 2013 #4
    Use Quotient Rule

    [itex]\frac{d}{dt}\left(\frac{u}{v}\right) = \frac{vu'-uv'}{v^2}[/itex]

    You should be able to solve this.
     
  6. Jul 14, 2013 #5

    tiny-tim

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    do you mean y1 = 1/(2) y0 ?

    yes you could start with that, then find y2, then y3, …

    and see if you can spot a pattern :smile:
     
  7. Jul 14, 2013 #6

    HallsofIvy

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    IF that is true then it is the answer to your problem! If you can start writing down the answer, you surely don't need our help! How did you get that?

    Poor, not so brilliant people like me might start by writing out a few values and looking for a pattern. If [itex]y_{n+1}= ((n+1)/(n+2))y_n[/itex], then [itex]y_1= ((0+1)/(0+2))y_0= y_0/2[/itex], [itex]y_2= ((1+1)/(1+2))y_1= 2y_1/3= 2(y_0/2)/3= y_0/3[/itex], [itex]y_3= ((2+1)/(2+2)y_2= (3/4)y_2= (3/4)(y_0/3)= y_0/4[/itex], [itex]y_4= ((3+1)/(3+2)y_3= (4/5)y_3= (4/5)(y_0/4)= y_0/5[/itex]...

    Do you think you can make a guess now? To be complete, you should then prove that your guess is correct, by induction, say.
     
  8. Jul 14, 2013 #7
    jhosamelly, I got 3/(n+2)^2 by quotient rule. How is that the answer?
     
  9. Jul 14, 2013 #8
    Thanks everyone.
     
  10. Jul 14, 2013 #9

    HallsofIvy

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    This problem has nothing to do with Calculus, the derivative, or the quotient rule.
     
  11. Jul 14, 2013 #10
    Ow, sorry, I thought your title is "solve the DIFFERENTIAL equation" .
     
  12. Jul 15, 2013 #11

    tiny-tim

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    spot the difference! :biggrin:
     
  13. Jul 21, 2013 #12

    HallsofIvy

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    But what you wrote had nothing to do with solving a differential equation either.
     
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