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Math Help - Proving These are Exponential

  1. #1
    Junior Member
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    Proving These are Exponential

    How would I go about proving that given

    f'(x)=f(x)-g(x)
    g'(x)=g(x)-f(x)

    Then f(x) and g(x) must be exponential functions.

    Thank you
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  2. #2
    Super Member
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    Differentiate the first equation to get f''(x) = f'(x) - g'(x) = 2f(x)

    And from there you should (probably) know how to solve it (since you're being asked this :P)
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  3. #3
    Math Engineering Student
    Krizalid's Avatar
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    the problem here is that we don't know if f is twice differentiable.

    here's a way to proceed: we have f'(x)-g'(x)=2\big(f(x)-g(x)\big), then \big(f(x)-g(x)\big)'=2\big(f(x)-g(x)\big), so f(x)-g(x)=c_1e^{2x}, where c_1\in\mathbb R.

    on the other hand is easy to see that f(x)+g(x)=c_2 for c_2\in\mathbb R, whereat f(x)=\dfrac{c_1e^{2x}+c_2}2 and g(x)=\dfrac{c_2-c_1e^{2x}}2, as required.
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