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Thread: Find the function

  1. #1
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    Find the function

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

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

    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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    Quote Originally Posted by qzno View Post
    Find a positive, continuous function f(x) which satisfies the integral equation

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

    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 $\displaystyle f'(x),$ 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 $\displaystyle f(x)$ 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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    $\displaystyle \frac{d}{dx} \left[ \int_a^x f(t) \, dt \right] = f(x)$
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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:

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

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

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

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

    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

    $\displaystyle \int \frac{1}{u} du$
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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

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

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

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

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

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

    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 $\displaystyle C\text.$ Just pick something that works.
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  13. #13
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    $\displaystyle f(x) = e^{3 \pi x + c_2} - c_1$

    or

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


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