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Math Help - What is this?

  1. #1
    Junior Member ReneG's Avatar
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    What is this?

    A friend of mine gave this to me as a challenge. I almost have no clue what I'm supposed to do

    he says "Solve for the following equation for f(x)"

    f(x) = x + \lambda \int_0^1 f(\xi) \,\,d \xi

    can anyone tell me what type of problem this is so I can do a bit of research on my own? My understanding of calculus only spans from Calc I to Calc II
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  2. #2
    MHF Contributor
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    Re: What is this?

    Hey ReneG.

    Hint: Try differentiating both sides to get a differential equation and solve from there.

    (These kinds of problems are known as integro-differential equations)

    Integro-differential equation - Wikipedia, the free encyclopedia
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  3. #3
    Junior Member ReneG's Avatar
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    Re: What is this?

    Thank you for pointing me in the right direction
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    Re: What is this?

    Hi ReneG,
    If the upper limit of your integral is x and not 1, I agree with Chiro. However as written, the integral is just a number c. So integrate both sides of your equation from 0 to 1 and get:

    c={1\over2}+\lambda c or c={1\over2(1-\lambda)}

    So f(x)=x+{\lambda\over2(1-\lambda)}
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  5. #5
    Junior Member ReneG's Avatar
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    Re: What is this?

    Quote Originally Posted by johng View Post
    Hi ReneG,
    If the upper limit of your integral is x and not 1, I agree with Chiro. However as written, the integral is just a number c. So integrate both sides of your equation from 0 to 1
    No typos. How did you integrate both sides though?

    \begin{align*} f(x) &= x + \lambda \int_{0}^{1}f(\xi)\, d\xi \\ \int_{0}^{1}f(x)\,dx &= \int_0^1 x \,dx + \lambda \int_{0}^{1} \left[ \int_{0}^{1} f(\xi)\,d\xi \right ] \,dx  \\  \int_{0}^{1}f(x)\,dx &= \frac{1}{2} + \lambda \int_{0}^{1} \left[ \int_{0}^{1} f(\xi)\,d\xi \right ]\,dx \end{align}

    I'm lost.
    Last edited by ReneG; July 2nd 2013 at 11:18 PM.
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  6. #6
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    Re: What is this?

    Hi again,
    It's easier for me to type the symbols in my own editor than to struggle with Latex and this HTML editor, so see the following "png" attachment:

    What is this?-mhfcalc12.png

    I hope this clears it up.
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  7. #7
    Junior Member ReneG's Avatar
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    Re: What is this?

    Quote Originally Posted by johng View Post
    I hope this clears it up.
    I appreciate it, thank you.
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