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2nd order DEQ, conserved quantity

  1. Mar 12, 2007 #1
    1. The problem statement, all variables and given/known data

    Given: y'' - y - (y^3) = 0 (equation 1)

    E = (1/2)(v^2) - (1/2)(y^2) - (1/4)(y^4) (equation 2)

    v = y'

    i. Show that E is a conserved quanitity
    ii. Find all the solutions with E = 0

    2. The attempt at a solution

    I'm not sure how to show a quantity is being conserved. In fact, I have no idea how to begin this problem!!!! Does anybody have some information to help me just get started?

    I solved equation 2 for v = (E + y^2 + (1/2)y^4)^(1/2). I realize I can integrate and solve for y(t) but it is really very messy and I don't see how doing so can immediately benefit me.
    Last edited: Mar 12, 2007
  2. jcsd
  3. Mar 12, 2007 #2


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    Science Advisor

    "Conserved" (as in conservation of energy and conservation of momentum) means "does not change". I.e. the derivative is 0.

    To show that E is a "conserved quantity", differentiate E (with respect to whatever the independent variable is) and use the given differential equation (y"= y+ y3) to show that the derivative is 0.
  4. Mar 12, 2007 #3
    Question: How do I differentiate E with respect to t when y and v are dependant upon t?

    here is my attempt...

    E = (1/2)(v^2) - (1/2)(y^2) - (1/4)(y^4) where v = (dy/dt) =>

    dE/dt = (dv/dt) - (dy/dt) - (dy/dt) = y'' - 2y'
    Last edited: Mar 12, 2007
  5. Mar 12, 2007 #4


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    Use the chain rule. For example, [tex]\frac{d}{dt}(y^2)=\frac{d}{dy}(y^2)\frac{dy}{dt}[/tex]
  6. Mar 12, 2007 #5
    thank you everyone; I have proven that E is conserved.
    Last edited: Mar 12, 2007
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