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TedG Group Title

anyone able to help me with difference equations?

  • one year ago
  • one year ago

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  1. vf321 Group Title
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    What specifically? I know a bit...

    • one year ago
  2. TedG Group Title
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    @vf321 sorry was watching the rugby :D. how do you take two recurrance equations ( i think thats correct term) and create a difference equation that relates the two. More specifically...

    • one year ago
  3. TedG Group Title
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    \[d _{n}=-2p_{n}+3 \] and \[s _{n+1}=p^{2}_{n}+1\] need a difference equation that relates \[p_{n+1}\] to \[p_{n}\]

    • one year ago
  4. vf321 Group Title
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    Well hang on-what is the relation between \(d_n\) and \(s_n\)?

    • one year ago
  5. TedG Group Title
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    you may need the information that P_{n} represents price per unit in period n. s_{n}, d_{n} represents supply and demand respectively. the question says asumming market price is price at which supply equals demand... i think that is answering what you have just asked.

    • one year ago
  6. vf321 Group Title
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    OK then. Well, then what's the problem?\[s_{n+1} = d_{n+1} = -2 p_{n+1}+3\]

    • one year ago
  7. TedG Group Title
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    but d_{n} is given, not d_{n+1}

    • one year ago
  8. vf321 Group Title
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    Yes, but that's the point of difference equations - they're true for all integral \(n > 0\)

    • one year ago
  9. vf321 Group Title
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    Or, depending on your definitions, \(n = 0\) may be valid too.

    • one year ago
  10. TedG Group Title
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    so, d_{n+1} is simply -2p_{n+1}+3

    • one year ago
  11. vf321 Group Title
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    yeah. Given that, can you replace \(s_{n+1}\) in the other equation?

    • one year ago
  12. vf321 Group Title
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    BTW, for inline latex surround your formula in \.(formula\.) ( without the periods)

    • one year ago
  13. TedG Group Title
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    so the difference equation would be -2p_{n+1}+3=p^{2}_{n}+1, and rearranged?

    • one year ago
  14. vf321 Group Title
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    Correct. Do you have Mathematica?

    • one year ago
  15. TedG Group Title
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    Mathematica, no. is it software?

    • one year ago
  16. vf321 Group Title
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    Yes, it would let you solve the difference equation if you needed to.

    • one year ago
  17. TedG Group Title
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    oh... I use maple. pretty sure that will help solve it. also thanks for the tip on the inline latex. been wandering how to do that for ages.

    • one year ago
  18. vf321 Group Title
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    Yup. np.

    • one year ago
  19. TedG Group Title
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    do you know about steady states? because thats what i am asked to find.

    • one year ago
  20. vf321 Group Title
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    Hmm. If I had to guess (so, no, I haven't really done them before), then I would say that steady states are situations where the price has asymptotic behavior, so as \(n\) goes up, \(p_n\) continually gets closer to a certain value.

    • one year ago
  21. TedG Group Title
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    hmm ok. Well I think the steady states are a diferrent way of saying stationary points. as we were taught them with an example using dx/dt=f(x*) and the steady states were solutions of f(x*)=0.

    • one year ago
  22. TedG Group Title
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    not sure if that will change the way you think about it.

    • one year ago
  23. TedG Group Title
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    oh ok. so do you have an idea on how you would approach finding steady states of a difference equation?

    • one year ago
  24. vf321 Group Title
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    Well, (and again I dont know for sure), being in a steady state for something discrete like a difference equation probably means solving \(p_{n+1}-p_{n} = 0\), because the difference between the two steps is 0 means it's in a steady state.

    • one year ago
  25. vf321 Group Title
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    And you already have a formula for \(p_{n+1}\), so I would just plug it in and solve.

    • one year ago
  26. TedG Group Title
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    oh ok. that makes sense. well anyway i guess that is everything. thanks for the help.

    • one year ago
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