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roadjester

  • one year ago

How the heck do I solve this diff. eq?

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  1. roadjester
    • one year ago
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    \[\large ({{y \over x} + e^{-xy}})dx + dy =0\]

  2. oldrin.bataku
    • one year ago
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    First, multiply through by \(xe^{xy}\):$$(ye^{xy}+x)dx+xe^{xy}dy=0$$Do you know how to solve from here?

  3. roadjester
    • one year ago
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    It doesn't look seperable, I still have y in the exponent of the dx term

  4. oldrin.bataku
    • one year ago
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    If only there were some function \(f\) s.t. \(\frac{\partial f}{\partial x}=ye^{xy}+x\), \(\frac{\partial f}{\partial y}=xe^{xy}\)... (I didn't solve it using separation of variables)

  5. roadjester
    • one year ago
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    you did partial differentiation...but I'm not quite sure of what

  6. oldrin.bataku
    • one year ago
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    Have you studied exact differential equations yet? If we integrate both sides of our original equation, assuming that there is some function \(f\) whose partial derivatives match whatever we've multiplied by \(dx\), \(dy\) respectively, we end up with \(f(x,y)=c\)... now, we merely need to determine what \(f\) is!

  7. roadjester
    • one year ago
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    not yet

  8. roadjester
    • one year ago
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    only homogenous diff. eq, and substitution

  9. oldrin.bataku
    • one year ago
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    Well, we know it's not homogeneous:$$(\frac{y}x+e^{-xy})dx+dy=0\\\frac{dy}{dx}=-[\frac{y}x+e^{-xy}]\\$$

  10. roadjester
    • one year ago
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    I get how you got that...but how does writing it as dy/dx help?

  11. oldrin.bataku
    • one year ago
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    It's to show how it's not homogeneous.

  12. roadjester
    • one year ago
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    how can you tell it's not homogeneous?

  13. roadjester
    • one year ago
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    \[\large y(1)=0\]

  14. oldrin.bataku
    • one year ago
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    Does \(\frac{ty}{tx}+e^{-(tx)(ty)}=\frac{y}x+e^{-xy}\)?

  15. roadjester
    • one year ago
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    I don't know? That was an Intial condition that I forgot to type...

  16. oldrin.bataku
    • one year ago
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    Do you know how to test whether an ODE is homogeneous?

  17. roadjester
    • one year ago
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    ODE? I'm guesing DE is diff. eq.

  18. oldrin.bataku
    • one year ago
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    ordinary differential equation

  19. roadjester
    • one year ago
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    no clue

  20. oldrin.bataku
    • one year ago
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    So you haven't learned about homogeneous differential equations?

  21. roadjester
    • one year ago
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    homogeneous yes, ordinary, not a term we've used

  22. oldrin.bataku
    • one year ago
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    Ordinary is on contrast with partial differential equations, where you have partial derivatives.

  23. roadjester
    • one year ago
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    you realize that I'm totally confused right? this is only my second class meeting

  24. oldrin.bataku
    • one year ago
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    :-p ok well tell me what you've been taught so far.

  25. roadjester
    • one year ago
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    just separating the terms and substitution if the equation can't be solved on the spot

  26. oldrin.bataku
    • one year ago
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    I don't think it can be solved using either technique. Maybe you need to learn how to solve exact equations first.

  27. ljensen
    • one year ago
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    Use the substitution \(u(x)=xy(x).\) Then \(u^\prime=xy^\prime+y\). This results in an easy equation for \(u(x).\)

  28. robtobey
    • one year ago
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    Hope this helps.

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  29. oldrin.bataku
    • one year ago
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    ... are you seriously just posting links to what Mathematica or Wolfram|Alpha spits out?

  30. robtobey
    • one year ago
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    Yes.

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