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Math Help - dy/dx = -y/x

  1. #1
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    dy/dx = -y/x

    Hello everybody,

    I need to solve

    dy/dx = -y/x

    for some economics problem. Can anybody help me?

    Thanks,

    Rob
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  2. #2
    MHF Contributor chisigma's Avatar
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    The DE you propose can be written as...

    \frac {dy}{y} = -\frac{dx}{x} (1)

    Al thet you have to do is to integrate first and second term of (1) indicating the 'arbitrary constant' as \ln c...

    Kind regards

    \chi \sigma
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  3. #3
    MHF Contributor
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    Quote Originally Posted by robvaneijk View Post
    Hello everybody,

    I need to solve

    dy/dx = -y/x

    for some economics problem. Can anybody help me?

    Thanks,

    Rob
    \frac{dy}{dx} = -\frac{y}{x}

    \frac{1}{y}\,\frac{dy}{dx} = -\frac{1}{x}

    \int{\frac{1}{y}\,\frac{dy}{dx}\,dx} = \int{-\frac{1}{x}\,dx}

    \int{\frac{1}{y}\,dy} = -\int{\frac{1}{x}\,dx}

    \ln{|y|} + C_1 = -\ln{|x|} + C_2

    \ln{|y|} + \ln{|x|} = C, where C = C_2 - C_1

    \ln{|y||x|} = C

    \ln{|xy|} = C

    |xy| = e^C

    xy = \pm e^C

    xy = A where A = \pm e^C

    y = \frac{A}{x}.
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