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Math Help - Partial Differential Equation

  1. #1
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    Partial Differential Equation

    Find the solution to the below partial differential equation

    d^2u/dx^2 =0
    d^2u/dydx =0
    d^2u/dy^2 =0

    the ans is:

    u(x,y) = Ax + By + C


    when I did my working I got:

    u(x,y)= Ax + g(y) where g(y) is an arbitary function.

    I just want to know iff these 2 results are equivelent or my result is wrong?
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  2. #2
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    Those are three different differential equations, not one.

    The general solution to [tex]\frac{\partial^2u}{\partial x^2}= 0[/itex] is u(x,y)= Ax+ g(y). But that does not necessarily satisfy the other equations. If you put u(x,y)= Ax+ g(y) into the third equation, you get [tex]\frac{d^2g}{dy^2}= 0[tex] which has general solution g(y)= By+ C. So u(x,y)= Ax+ By+ C. You can then check to see that it also satisfies the second equation: \frac{\partial^2(Ax+ By+ C)}{\partial y\partial x}= \frac{\partial A}{\partial y}= 0.
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