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Math Help - solving 2nd order PDE

  1. #1
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    solving 2nd order PDE

    Need to solve for dx/ds in the following equation, keeping dy/ds.

    d^2x/ds^2 - (2/y)(dx/ds)(dy/ds) = 0

    I can just rearrange to get:

    dx/ds = (y/2)(ds/dy)(d^2x/ds^2)

    But, this is not clean to use for some later calculations.

    Is there any way to solve for dx/ds by integration?I would like an expression with dx/ds = f(y,s)
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  2. #2
    MHF Contributor
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    First off, this is and ODE since there are only ordinary derivatives here. If you write your ODE as

    \dfrac{\dfrac{d^2x}{ds^2}}{\dfrac{d x}{d s} }= 2 \dfrac{\dfrac{dy}{ds}}{y}

    You can integrate giving

    \ln \dfrac{d x}{d s}} = 2 \ln y + \ln c

    or

    \dfrac{dx}{ds} = c y^2
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