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What guess should I use when the right hand side is a constant?

  1. Dec 10, 2012 #1
    What form should the particular solution of a 2nd order linear non homogeneous differential equation take when the right hand side is a constant?

    if the differential differential equation has the form

    Ay''+By'+Cy = g(x)

    where g(x) is a constant

    what form should the particular solution take?

    I know if g(x) is an exponential the "trial" solution should be A(e^x)

    If g(x) is trigonometric it should be A(sin(x))+B(cos(x))

    And If it is a polynomial of degree n it should be A(x^n) + B(x^(n-1)) ... +C(x^0)

    But what if the differential equation is just Ay''+By'+Cy = D

    Should the trial solution be a polynomial of degree 0?

    So I would try
    y = D
    y' = 0
    y'' = 0

    which would leave me with

    CD=D

    which is useless

    Please Help!

    Thanks!
     
  2. jcsd
  3. Dec 10, 2012 #2

    mfb

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    2016 Award

    Staff: Mentor

    Well, ##y=\frac{D}{C}## is a solution to your equation. It is not the only one, so you can add all solutions to Ay''+By'+Cy = 0. You can use both the exponential and the trigonometric approach (with complex numbers, they are the same anyway) to find them.
     
  4. Dec 11, 2012 #3
    Ahhh okay, thank you for your answer :D
     
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