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Question on Ordinary Differential Equation (ODE)

  1. Aug 25, 2013 #1
    1. The problem statement, all variables and given/known data

    Find the ODE of the following
    (1) du/dy = -u
    (2) d^2u/dxdy = -du/dx


    2. Relevant equations

    For question 1, the answer is u= A(x)e^(-y)
    while for question 2, the answer is u= e^(-y)(B(X) + c(Y))


    3. The attempt at a solution

    I've already solved the question, but just want to ask, if i change (1) to (2), its just a differentiation with respect to dx, but yet the solution contain a c(Y) term, any idea why is this so?
     
  2. jcsd
  3. Aug 25, 2013 #2

    haruspex

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    To get those answers, these are PDEs, surely?
    In (2), both sides have been differentiated partially wrt x. So any additive term that depends solely on y will have disappeared and cannot be reconstructed from the PDE. It's analogous to the constant of integration in ODEs.
     
  4. Aug 25, 2013 #3

    CAF123

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    If you differentiate (1) wrt x, then
    d/dx (du/dy) = d/dy (du/dy) dy/dx = d2u/dy2 dy/dx, by chain rule.
     
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