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Thread: True/False Proof

  1. #1
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    True/False Proof

    True or false: If $\displaystyle f(x)$ is a solution to $\displaystyle \frac{dy}{dx}=2x$, then $\displaystyle f'(x)$ is a solution to $\displaystyle \frac{dy}{dx}=2y$. Justify your answer.


    I have no idea how to go about this...
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  2. #2
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    Quote Originally Posted by jangalinn View Post
    True or false: If $\displaystyle f(x)$ is a solution to $\displaystyle \frac{dy}{dx}=2x$, then $\displaystyle f'(x)$ is a solution to $\displaystyle \frac{dy}{dx}=2y$. Justify your answer.


    I have no idea how to go about this...
    I assume you're letting $\displaystyle y = f(x)$ and $\displaystyle \frac{dy}{dx} = f'(x)$.


    $\displaystyle \frac{dy}{dx} = 2x$

    $\displaystyle y = \int{2x\,dx}$

    $\displaystyle = x^2 + C$.


    So $\displaystyle y = f(x) = x^2 + c$.


    Now since $\displaystyle y = x^2 + c$

    $\displaystyle x^2 = y - c$

    $\displaystyle x = (y - c)^{\frac{1}{2}}$

    $\displaystyle \frac{dx}{dy} = \frac{1}{2}(y - c)^{-\frac{1}{2}}$

    $\displaystyle \frac{dx}{dy} = \frac{1}{2(y - c)^{\frac{1}{2}}}$

    $\displaystyle \frac{dy}{dx} = 2(y - c)$

    $\displaystyle \frac{dy}{dx} = 2y - 2c$

    $\displaystyle f'(x) = 2y + C$ where $\displaystyle C = -2c$.


    So the statement is true.
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