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Math Help - Partial differential equation

  1. #1
    Newbie mukmar's Avatar
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    Partial differential equation

    I'm trying to remember how to solve a type of equation I've reduced a second order partial differential to.

    In the notes it goes from the equation:
    2\xi u_{\xi\eta} - u_{\eta} = 0

    to
    <br />
2\xi u_{\xi\eta} - u_{\eta} = 2\xi^{3/2}\frac{d}{d\xi}\left(\frac{u_{\eta}}{\xi&{1/2}}\right) = 0

    With subscripts denoting partial derivatives.

    I know that these two steps are equivalent, I'm just wondering on what the procedure used is to get the second equivalent form.

    If the procedure has a name which I can look up.

    Any assistance would be greatly appreciated, thank you.
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  2. #2
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    Remove \displaystyle 2\xi^{3/2} as a 'common factor' and see that the thing inside the brackets is a perfect derivative.

    Alternatively, use the integrating factor technique used for solving some first order ode's.
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  3. #3
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    Quote Originally Posted by mukmar View Post
    I'm trying to remember how to solve a type of equation I've reduced a second order partial differential to.

    In the notes it goes from the equation:
    2\xi u_{\xi\eta} - u_{\eta} = 0
    to
    <br />
2\xi u_{\xi\eta} - u_{\eta} = 2\xi^{3/2}\frac{d}{d\xi}\left(\frac{u_{\eta}}{\xi&{1/2}}\right) = 0

    With subscripts denoting partial derivatives.

    I know that these two steps are equivalent, I'm just wondering on what the procedure used is to get the second equivalent form.

    If the procedure has a name which I can look up.

    Any assistance would be greatly appreciated, thank you.
    It might be easier to let u_{\eta} = v so you have 2 \xi v_{\xi} = v a separable ODE.
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