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Solving tricky functional equation

  1. Jul 15, 2014 #1
    Consider the following linear functional operator:

    $$Q_w[f(x)] = \lim_{h\rightarrow w} \lbrace \frac{f(x + h) - f(x)}{h} \rbrace $$

    How does one solve the equation

    $$a_0(x)Q_0[f(x)] = a_1(x)Q_1[f(x)]$$

    Spelt out that is:

    $$a_0(x)*f'(x) = a_1(x)(f(x+1) - f(x))$$

    For the case of constant functions $a_0(x) = a_0$ and $a_1(x) = a_1$ the solution is simply found by assuming

    $$f(x) =e^{Lx}$$

    thereby implying:

    $$a_0L e^{Lx} = a_1(e^{Lx}(e^{L} - 1))$$

    which can be solved as

    $$\frac{L}{1 - e^{L}} = =\frac{a_1}{a_0} $$

    And L can be extracted through the use of Lambert-W function.

    But what about more general functions?
  2. jcsd
  3. Jul 16, 2014 #2
    This is a delay differential equation, so I'd start looking there. (Not my expertise unfortunately)
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