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Nonlinear integration using integration factors?

  1. Jul 25, 2011 #1
    Hi,

    I have some equations I have derived though energy balances and have been using the Integration Factors method to solve equations in the form:

    dy/dx + Py = Q

    However, my original equations only work if I assume some variables to be constants. Removing this assumption leaves me with an equation of the following form:

    dy/dx + (a+b.y^2)y = Q

    I have no Idea how to solve this problem. I cannot find a method that works. Does anybody know how to integrate such a problem?

    Thanks in advance
     
  2. jcsd
  3. Jul 25, 2011 #2

    hunt_mat

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    This looks like a Bernoulli type possibly.
     
  4. Jul 25, 2011 #3
    Thanks for the reply matt.

    I have looked at the Bernoulli's method but as far as I am aware Bernoulli only works for linear integrations. Please advise me if I'm wrong and if possible, where I can see further information.
     
  5. Jul 25, 2011 #4
    I'm not sure this is right, but it may work!:

    dy/dx + (a+b.y^2)y = Q

    dy/dx + ay+by^3 = Q

    dy/dx = Q - ay - by^3

    y = Qx - a[itex]\int(ydx[/itex]-b[itex]\int(y^3dx[/itex]

    Now we convert dx into dy and y terms:

    dy/dx = Q - ay - by^3

    dx = [itex]\frac{dy}{Q - ay - by^3}[/itex]

    Bam! Here it is

    y = Qx - a[itex]\int(\frac{y}{Q - ay - by^3}dy[/itex]-b[itex]\int(\frac{y^3}{Q - ay - by^3}dy[/itex]
     
  6. Jul 25, 2011 #5
    The above method is too tedious, here is a better way:

    dy/dx = Q - ay - by^3

    dx/dy = [itex]\frac{1}{Q - ay - by^3}[/itex]

    dx = [itex]\frac{dy}{Q - ay - by^3}[/itex]

    Now integrate both sides (I don't know how to do it) but Wolf math does:

    http://integrals.wolfram.com/index.jsp?expr=1/(c+-+ax+-+bx^3)&random=false

    Does anyone understand it's answer though?
     
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