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Math Help - Separation of variables

  1. #1
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    Unhappy Separation of variables

    hi all,

    im practising separation of variables for partial differential equations. i came across this site which says:

    2. Separation of Variables


    'NOTE: In this variables separable section we only deal with first order, first degree
    differential equations.'

    so , if i have a second order or above PDE, then i cant use separation of variables?
    i thought one can use separation of variables for example in the wave equation!! that is a second order PDE....

    ok im confused
    can anyone help me out?
    i hope i am clear with this question problem

    thank you all very much
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  2. #2
    MHF Contributor

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    I think you and the site are talking about different things.

    You are talking about separating independent variables in a partial differential equation:
    i.e. solving \frac{\partial^2 \phi}{\partial x^2}= \frac{\partial^2\phi}{\partial t^2} by writing \phi(x, t)= X(x)T(t), the product of two functions, each a function of one of the independent variables.

    The site you cite is talking about separating the independent and dependent variables in an ordinary differential equation:
    \frac{dy}{dx}= xy- x^2y= y(x- x^2)
    which can be written as \frac{dy}{y}= (x- x^2)dx.

    That latter can only be done with first order equations.
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  3. #3
    MHF Contributor chisigma's Avatar
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    For certain types of second order DE You can use separation of variables. An example is...

    y^{''} = f(y) (1)

    With the substitution...

    \displaystyle y^{''} = \frac{d y^{'}}{dy} \ \frac{d y}{dx} = y^{'}\ \frac{d y^{'}}{dy} (2)

    ... the (1) becomes the first order separate variables DE...

    \displaystyle y^{'}\ \frac{d y^{'}}{dy}= f(y) (3)

    ... the solution of which is...

    \displaystyle y^{'} = \sqrt {2\ \phi (y) + c_{1}} (4)


    ... where \phi(y) is a primitive of f(y). Now if in (4) You separe the variables again You obtain...

    \displaystyle dx = \frac{dy}{\sqrt {2\ \phi (y) + c_{1}}} (5)

    ... so that the general solution of (1) is...

    \displaystyle x= \int \frac{dy}{\sqrt {2\ \phi (y) + c_{1}}} + c_{2} (6)

    Kind regards

    \chi \sigma
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