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Bernoulli equation

  1. Oct 8, 2005 #1
    can someone explain how to solve the bernoulli equation? i'm having a hard time understanding...
     
  2. jcsd
  3. Oct 8, 2005 #2

    mezarashi

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    The Bernoulli equation is an energy conservation equation for fluid kinetics. In what way are you having difficulty solving it? o_O
     
  4. Oct 8, 2005 #3

    saltydog

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    You mean:

    [tex]y^{'}+P(x)y=Q(x)y^n[/tex]

    The key to solving this is to recognize the differential form:


    [tex]y^{-n}dy[/tex]

    and what, when differentiated, gives this. Well that would be:

    [tex]\frac{1}{1-n}y^{1-n}[/tex]

    Hey, I know it's not easy. They catch me in here all the time with differential forms.

    Ok then so we'll divide by [itex]y^n[/itex] up there in the first equation and take the differential form:

    [tex]y^{-n}dy+Py^{1-n}dx=Qdx[/tex]

    Alright then,so that's what we have right, the differential [itex]y^{-n}dy[/itex].

    So, let:

    [tex]z=y^{1-n}[/tex]

    and then substitute the differential form of this into the original equation. Here's the first part:

    We got:

    [tex]y^{-n}dy+Py^{1-n}dx=Qdx[/tex]


    So the [itex]y^{-n}dy[/itex] part would just be:

    [tex]\frac{1}{1-n}dz[/tex]

    Do the rest and then get a first-order ODE in terms of z and x.
     
    Last edited: Oct 8, 2005
  5. Oct 8, 2005 #4
    hmm... so the key is to try to get the non-linear equation into a linear equation...
    saltydog, thank you very much for explaining it to me!!! :)
     
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