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Thread: Functional equationsolving find f(x)

  1. #1
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    Functional equationsolving find f(x)

    Hi, I am trying to solve the following functional equation

    f(x)-f(2a-x)=b(x-a)

    considering "a" and "b" are positive constants.
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  2. #2
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    Re: Functional equationsolving find f(x)

    there are two parameters so let's try a function of the form

    $f(x) = c_1 x + c_0$

    $c_1 x + c_0 - (c_1(2a-x)+c_0) = b(x-a)$

    $2c_1 x - 2 a c_1 = b x - a b$

    $2c_1 = b$

    $c_1 = \dfrac b 2$

    $-2ac_1 = -a b$

    $a$ is specified positive so

    $2c_1 = b$

    $c_1 = \dfrac b 2$

    and we have agreement in the value of $c_1$

    so we have

    $f(x) = \dfrac b 2 x + c_0$

    There are of course higher order polynomials that satisfy this functional relationship but this is the lowest order one.
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    Re: Functional equationsolving find f(x)

    Thanks romsek. I knew that, the second order polynomial also satisfies, but not higher order ones.
    However to solve the functional equation we need to find all functions, not just examples that satisfy.
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    Re: Functional equationsolving find f(x)

    Quote Originally Posted by Rafael View Post
    Thanks romsek. I knew that, the second order polynomial also satisfies, but not higher order ones.
    However to solve the functional equation we need to find all functions, not just examples that satisfy.
    what kind of answer is acceptable then? I doubt there's going to be some notation that is all encompassing of the various functions that satisfy this.

    Are you supposed to list them all out?
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  5. #5
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    Re: Functional equationsolving find f(x)

    Well, the first order polynomial is the special case of the second order polynomial, but to solve the functional equation one should give a complete solution, maybe there is a more general function that becomes, for example, second order polynomial in the special case but still satisfies the equaion.

    for example, let's solve following functional equation:

    f(x)-2f(1/x)=2^x

    we can replace x by 1/x

    f(1/x)-2f(x)=2^(1/x)

    from these two equations, one can determine that f(x) =-(1/3)(2^x+2^(1-1/x))
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    Re: Functional equationsolving find f(x)

    Quote Originally Posted by Rafael View Post
    Hi, I am trying to solve the following functional equation

    f(x)-f(2a-x)=b(x-a)

    considering "a" and "b" are positive constants.
    If we take $x = a$ in the given equation we have $f(a)-f(a) = 0$ so $f(a)$ is arbitrary. Let's call $f(a)=C$. Now let's assume $f$ is differentiable and differentiate the equation giving
    $f'(x) + f'(2a-x) = b$ and put $x=a$ in that. That gives $f'(a) + f'(a) = b$ so $f'(a) = \frac b 2$. Let's differentiate again:
    $f''(x) - f''(2a-x) = 0$. Putting $x=a$ in that gives $f''(a)-f''(a) = 0$, so $f''(a)$ is arbitrary. Lets call $f''(a) = D$. Let's differentiate again:
    $f'''(x) + f'''(a-x) = 0$. Putting $x=a$ in that gives $f'''(a)+f'''(a) = 0$, so $f'''(a)$ is $0$.
    Now a pattern becomes clear. From now on all the odd derivatives will be $0$ and the even ones will be arbitrary when $x=a$. So let's look at the Taylor series
    for $f(x)$ about $x=a$.
    $$f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!}(x-a)^k = C + \frac b 2 (x-a) + \frac D 2 (x-a)^2 +\text{more terms with even powers of }(x-a)$$
    Notice that if we take the arbitrary constant $D=0$ we get $f(x) = C + \frac b 2 (x-a) = C-\frac{ab} 2 + \frac b 2 x$ which agrees with the earlier first degree
    solution. And it satisfies the functional equation. So what about all those arbitrary terms of the form $K(x-a)^{2k}$. Notice that such expressions satisfy $f(x)-f(2a-x)=0$ so they don't disturb the functional relationship that is already satisfied by the first two terms. So the functional equation can be satisfied by adding any convergent sum of the form$$
    \sum_{k=1}^\infty a_k (x-a)^{2k}$$ to the first two terms.
    In fact, there is no real reason not to include the constant term as one of the even powers and write the answer as
    $$f(x) = \frac b 2 (x-a) + \sum_{k=0}^\infty a_k (x-a)^{2k}$$
    So, for example, a non polynomial example is $f(x) = \frac b 2(x-a) + \cos(x-a)$.
    Last edited by Walagaster; May 26th 2018 at 04:15 PM.
    Thanks from Rafael
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