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Differential equation: 1st order

  1. Oct 1, 2006 #1
    is there an exact solution to

    dy/dx = -x - y

    i am doing a modelling, and just happen to get stumbled into this form of pde.

    i do it numerically, but i also want to know how y behave as x approaches large value. i just need to present some analytical work to justify what happen as x grows large.

    thanks.
     
  2. jcsd
  3. Oct 1, 2006 #2

    quasar987

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    The EDO is linear the exact solution is

    [tex]y(x)=\frac{-\int x e^x dx}{e^x}[/tex]
     
  4. Oct 1, 2006 #3

    quasar987

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    P.S.

    [tex]\int xe^xdx = e^x(x-1)[/tex]

    So y=1-x
     
  5. Oct 4, 2006 #4

    HallsofIvy

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    Actually, the general solution is y= Ce-x+ 1- x. You forgot to include the constant of integration.
     
  6. Oct 4, 2006 #5
    Solve the homogeneous equation [tex]y' + y = 0[/tex], then consider [tex]y = Ax+B[/tex] as the particular solution. When you've got [tex]\mathcal{L}y = f(x)[/tex] where f(x) is an n'th order polynomial, trying [tex]y = a_{n}x^{n} + ... + a_{0}[/tex] gives the particular solution.
     
  7. Oct 4, 2006 #6

    HallsofIvy

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    That's one way to do it. Since this is a first order equation, it's also easy to find an integrating factor, which is what quasar987 did.
     
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