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Functional Equation?

  1. Jun 18, 2009 #1
    Hi All,

    I am asked to produce a function such that, literally, increasing the indipendent variable by lambda will produce an increase in the function of a*lambda.

    I thought about setting up an equation as follows

    y(lambda*x)=a*lambda*y(x)

    In general a simple solution of the functional equation y(ax)-by(x)=0 is y = Kx^(ln(a)/ln(b)). C is arbitrary

    Using this solution scheme I am unable to obtain solutions independent upon lambda.

    Is the task well posed at all?

    Thank you very much

    Muzialis
     
  2. jcsd
  3. Jun 18, 2009 #2

    EnumaElish

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    I would have formulated it as:

    Find y such that y(x + λ) = y(x) + aλ.
     
  4. Jun 18, 2009 #3
    EnumaElish,

    thanks for the hint.

    Your set up is coherent with my description.
    However I have been imprecise, the increase is specified in the task as a multiplicative factor, so my set up is the one actually that needs solving.

    I am pretty sure the task was badly posed though as I do not see how a solution could exist.

    But then, that is why I posted the issue.

    Thanks and all the best

    Muzialis
     
  5. Jun 18, 2009 #4

    EnumaElish

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    Could it mean y((1+λ)x) = (1+aλ)y(x)?

    With y(λx) = aλy(x), assume (trivially) λ = 1. Then y(x) = ay(x), which is a contradiction except for a = 1. In this formulation λ has to be > 1.
     
  6. Jun 18, 2009 #5
    EnumaElish,

    I appreciate your ad absurdum reasoning. But can you find any solution for lambda > 1?

    I am sure the task was badly posed. The fact is it was proposed to me by a working partner whose maths is usually very precise, so I wanted to be extra sure I was not stating nonsense.

    It has been later clarified that all it was needed was a function with a derivative of lambda, locally in a point of interest.

    I am curious now tough, if the set up I originally proposed has any solution at all, although I fear not.

    Thank you very much

    Best Regards

    Muzialis
     
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