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Math Help - True/False Proof

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    True/False Proof

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


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


    \frac{dy}{dx} = 2x

    y = \int{2x\,dx}

     = x^2 + C.


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


    Now since y = x^2 + c

    x^2 = y - c

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

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

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

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

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

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


    So the statement is true.
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