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Thread: differential

  1. #1
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    differential

    f'(x)=1/2f(x)-4
    Find f(x).

    It is easy to estimate f(x).
    It must be squarerootx+2. I found the result by experience. What operations can I do?
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  2. #2
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    Re: differential

    This is known as a differential equation. $\dfrac{d}{dx}\left(\sqrt{x}+2\right) = \dfrac{1}{2\sqrt{x}}$, so this does NOT satisfy your equation.

    Instead, you wind up with $f(x) = c_1 e^{x/2}+8$ where $c_1$ is any arbitrary constant.
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  3. #3
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    Re: differential

    f'(x)=1/(2f(x)-4) (maybe if I write like this it will be better.)
    Find f(x).

    Answer is :square root2+2. But I don't know a way to solve this equation. I just made an estimation.
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  4. #4
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    Re: differential

    Quote Originally Posted by kastamonu View Post
    f'(x)=1/(2f(x)-4) (maybe if I write like this it will be better.)
    Find f(x).

    Answer is :square root2+2. But I don't know a way to solve this equation. I just made an estimation.
    The correct answer is $f(x) = c_1e^{x/2}+8$. I just told you that. $\sqrt{2}+2$ is a constant. If $f(x) = \sqrt{2}+2$, then $f'(x) = 0$, and $0 \neq \dfrac{1}{2}(\sqrt{2}+2)-4$, so you are still wrong.
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  5. #5
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    Re: differential

    squarerootx+2
    I made a typo.
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  6. #6
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    Re: differential

    Slip - you're misreading the problem. You are solving this:

     f'(x) = \frac 1 2 (f(x)-4)

    But the OP meant this:

     f'(x) = \frac 1 {2 f(x)-4}

    Starting with:

     \frac {dy}{dx} = \frac 1 {2y -4}

    Rearrange and you get to:

     (2y-4)dy = dx

    Integrate:

     y^2-4y-(x+c) = 0

    where c is any arbitrary constant. This then yields:

     y = 2 \pm \sqrt{x+c} .

    So yes,  2 + \sqrt x is a solution, but so is 2 - \sqrt {x} , as well as  2 \pm  \sqrt {x+5}, etc.
    Last edited by ChipB; Jun 8th 2017 at 12:25 PM.
    Thanks from SlipEternal
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  7. #7
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    Re: differential

    Quote Originally Posted by kastamonu View Post
    (maybe if I write like this it will be better.)
    It would be better if you learn LaTeX.
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  8. #8
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    Re: differential

    Many Thanks.
    What is La Tex
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  9. #9
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    Re: differential

    Quote Originally Posted by kastamonu View Post
    Many Thanks.
    What is La Tex
    LaTex Tutorial

    It is how we are formatting math equations to be readable on these forums.
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