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Math Help - Find the function

  1. #1
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    Find the function

    Find a positive, continuous function f(x) which satisfies the integral equation

    f(x) = \pi \left( 2 + \int_{1}^{x} 3f(t) dt \right)

    Hint: start by differentiating each side.



    I Have Absolutely NO Idea On How To Do This Problem, I Don't Even Understand The Hint.
    Thanks To Anyone Who Helps <3
    : )
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  2. #2
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    Quote Originally Posted by qzno View Post
    Find a positive, continuous function f(x) which satisfies the integral equation

    f(x) = \pi \left( 2 + \int_{1}^{x} 3f(t) dt \right)

    Hint: start by differentiating each side.



    I Have Absolutely NO Idea On How To Do This Problem, I Don't Even Understand The Hint.
    Thanks To Anyone Who Helps <3
    : )
    Differentiate both sides of the equation, as it says. The left side will be f'(x), and for the right side, you will need to use the second part of the fundamental theorem of calculus. After that, you should be able to determine a suitable f(x) through integration.
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  3. #3
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    i still cant figure out how to differentiate the right side : (
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  4. #4
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    \frac{d}{dx} \left[ \int_a^x f(t) \, dt \right] = f(x)
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  5. #5
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    i know that : (
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  6. #6
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    Quote Originally Posted by qzno View Post
    i know that : (
    Okay, but we aren't mind readers. You need to tell us where you are getting stuck. Can you at least get the derivative? You have a constant (derivative is zero) plus an integral with a variable limit which can be differentiated using the property that skeeter gave; that part should be straightforward.

    Show us your attempts so far, and we can guide you in the right direction.
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  7. #7
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    I Got It Down To The Following:

    \int \frac{f'(x)}{f(x)} dx = \int 3\pi dx

    What do I do from here : )
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  8. #8
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    Quote Originally Posted by qzno View Post
    I Got It Down To The Following:

    \int \frac{f'(x)}{f(x)} dx = \int 3\pi dx

    What do I do from here : )
    Good. The right side becomes 3\pi x + C\text. Now what is \int\frac{u'}u\,dx?

    Apply the log rule, and then exponentiate both sides.
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  9. #9
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    can you do u substitution and do like

    u = f(x)
    du = f'(x) dx

    and then youd get

    \int \frac{1}{u} du
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  10. #10
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    Quote Originally Posted by qzno View Post
    can you do u substitution and do like

    u = f(x)
    du = f'(x) dx

    and then youd get

    \int \frac{1}{u} du
    Correct! You've got it.
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  11. #11
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    so i got:

    ln (f(x)) + C = 3 \pi x + C

    is that all i do?
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    Quote Originally Posted by qzno View Post
    so i got:

    ln (f(x)) + C = 3 \pi x + C
    Good, but the two constants of integration are not necessarily equal; you should either use separate names (like C_0 and C_1), or combine them as one constant.

    is that all i do?
    No. You want to find f(x) (i.e., solve for f(x)).

    Note that in this problem, you want to find a function that satisfies the conditions. There are many choices, depending on what you choose for C\text. Just pick something that works.
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  13. #13
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    f(x) = e^{3 \pi x + c_2} - c_1

    or

    f(x) = e^{3 \pi x + c_2 - c_1}


    Are either of these correct?
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  14. #14
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    Quote Originally Posted by qzno View Post
    f(x) = e^{3 \pi x + c_2 - c_1}
    Good. Now choose c_1 and c_2 so that f satisfies the original equation.
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  15. #15
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    how do u choose c_1 and c_2 ?
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