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Solve Linear ODE Simplify Step

  1. Sep 8, 2013 #1

    1s1

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    1. The problem statement, all variables and given/known data

    Solve: ##x\frac{dy}{dx}-4y=x^{6}e^{x}##

    2. Relevant equations

    ##x^{-4}\frac{dy}{dx}-4x^{-5}y=xe^{x}## is equal to ##\frac{d}{dx}[x^{-4}y]=xe^x##

    3. The attempt at a solution

    The second equation above simplifies to the third (according to my textbook) but I can't figure out how. Any help would be greatly appreciated!
     
    Last edited: Sep 8, 2013
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  3. Sep 8, 2013 #2

    vela

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  4. Sep 8, 2013 #3

    1s1

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    Thanks!
     
  5. Sep 8, 2013 #4

    vela

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    Look up "integrating factor." That's probably what the book is just about to show you how to find.

    In this particular case, the first equation divided by x5 gives you the second equation, right?
     
    Last edited: Sep 8, 2013
  6. Sep 8, 2013 #5

    1s1

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    Yes, the second equation is just the first divided by x5. (Which is the first put in standard form by dividing through by x and then multiplying by the integrating factor which is x-4 - so divided by x5 in total.)

    The book then makes the jump that:

    ##x^{-4}\frac{dy}{dx}-4x^{-5}y=xe^{x}## equals ##\frac{d}{dx}[x^{-4}y]=xe^{x}##

    and I can't figure out how this jump is made.
     
  7. Sep 8, 2013 #6

    LCKurtz

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    What do you get if you differentiate ##x^{-4}y## with respect to ##x## using the product rule?
     
  8. Sep 8, 2013 #7

    1s1

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    That's it! Thanks!
     
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