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Solve this system of DE's for the First Integral

  1. Oct 25, 2013 #1
    I am trying to solve this system DE's to determine the systems First Integral.

    dx/dt = y+x2-y2
    dy/dt = -x-2xy

    I am pretty sure I need to pick some different variables to use to make the equation easier to solve, but I can't get anything to work. I thought about letting a variable be x2y, but that doesn't help much. If anyone can help me that would be much appreciated!!
     
  2. jcsd
  3. Oct 25, 2013 #2

    tiny-tim

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    Hi LBJking123! :smile:

    Hint: suppose it was

    dx/dt = y+x2+y2
    dy/dt = x+2xy​

    what would you do? :wink:
     
  4. Oct 25, 2013 #3
    I tried solving by separation of variables, but I cant figure out how to get all of the x's to one side and y's to the other. I think I am totally missing something obvious....
     
  5. Oct 26, 2013 #4

    epenguin

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    As someone said, you need to pick some different variables. Can you do that for tiny-tim's example?
     
  6. Oct 26, 2013 #5

    tiny-tim

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    Yes, separation of variables won't work.

    Hint: suppose it was

    dx/dt = y
    dy/dt = x ?​
     
  7. Oct 27, 2013 #6
    That case you could divide the two equations, and then get xdx=ydy. Then I would integrate both sides to get the answer. That technique wont work for the original DE's though...
     
  8. Oct 28, 2013 #7

    tiny-tim

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    (just got up :zzz:)
    no, all that gives you is x2 - y2 = constant …

    how does that help? :redface:

    try that example again :smile:
     
  9. Oct 28, 2013 #8

    arildno

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    I am not QUITE sure where tiny-tim is trying to lead you. LBJking123, but I, at least, felt that the variable change u=x+y and v=x-y simplifies the equations in a manner that may be amenable for further simplifications.
     
  10. Oct 28, 2013 #9

    tiny-tim

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    hi arildno! :smile:

    yes, that would be the way to solve my easy example (but i was hoping LBJking123 would see it on on his own :redface:)
     
  11. Oct 28, 2013 #10

    arildno

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    Well, I spotted some further troubles on the way (I was hoping a g(u/v) substitution would turn up, but it doesn't seem to be THAT simple..)
    So, I have been following this thread for a while, and am hoping to see some real cleverness on your part in the end that I have missed.
    :smile:
     
  12. Oct 28, 2013 #11

    tiny-tim

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    think laterally! :wink:

    (but don't give away the answer)
     
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