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Math Help - Seperable Equations - Word Problem. Help Please.

  1. #1
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    Seperable Equations - Word Problem. Help Please.

    I need help with this question below. I don't really understand the problem. I tried drawing a diagram, but it still didn't help me much. I'm especially confused on the part where it states that "S satisfies a first-order differential equation", and how do I "solve it"?

    Any help will be truly appreciated.

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  2. #2
    MHF Contributor chisigma's Avatar
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    The equation in T is \dots

    \frac{d^{2}T}{dr^{2}}+\frac{2}{r} \frac{dT}{dr}=0; T(1)=15,T(2)=25 (1)

    Setting \frac{dT}{dr}=S the (1) becomes…

    \frac {dS}{dr}=-\frac {2}{r} \cdot S (2)

    … which can be written as \dots

    \frac {dS}{S}= - \frac {2}{r} \cdot dr (3)

    Integrating both terms of (3) we have \dots

    \ln S = - \int \frac {2}{r} \cdot dr = -2 \cdot \ln|r| + \ln c_{1} \rightarrow S= \frac {c_{1}} {r^{2}} (4)

    Further integration gives us \dots

    T= \int \frac {c_{1}}{r^{2}}\cdot dr + c_{2}= - \frac {c_{1}}{2\cdot r} + c_{2} (5)

    The conditions given in (1) permit us to find c_{1}=40, c_{2}=35, so that the solution is…

    T=35 - \frac{20}{r} (6)

    Kind regards

    \chi \sigma
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