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roadjester

  • 2 years ago

How the heck do I solve this diff. eq?

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

  2. oldrin.bataku
    • 2 years 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
    • 2 years ago
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    It doesn't look seperable, I still have y in the exponent of the dx term

  4. oldrin.bataku
    • 2 years 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
    • 2 years ago
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    you did partial differentiation...but I'm not quite sure of what

  6. oldrin.bataku
    • 2 years 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
    • 2 years ago
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    not yet

  8. roadjester
    • 2 years ago
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    only homogenous diff. eq, and substitution

  9. oldrin.bataku
    • 2 years 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
    • 2 years ago
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    I get how you got that...but how does writing it as dy/dx help?

  11. oldrin.bataku
    • 2 years ago
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    It's to show how it's not homogeneous.

  12. roadjester
    • 2 years ago
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    how can you tell it's not homogeneous?

  13. roadjester
    • 2 years ago
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    \[\large y(1)=0\]

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

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

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

  17. roadjester
    • 2 years ago
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    ODE? I'm guesing DE is diff. eq.

  18. oldrin.bataku
    • 2 years ago
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    ordinary differential equation

  19. roadjester
    • 2 years ago
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    no clue

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

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

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

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

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

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

  26. oldrin.bataku
    • 2 years 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
    • 2 years 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
    • 2 years ago
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    Hope this helps.

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

  30. robtobey
    • 2 years ago
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    Yes.

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