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2nd order non-homogeneneous ODE - how to find PI

  1. Dec 6, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the general and, if possible, particular solutions of the following ordinary differential equations:

    y''+9y=36sin3x
    (hint: modification rule for PI)

    2. Relevant equations
    Knowledge of ODE's
    [tex]y = y_{aux}+y_{particular}[/tex]

    3. The attempt at a solution
    I get the compementary function;
    y''+9y = 0

    and then using lambda-notation

    [tex]\lambda^2 + 9 = 0[/tex]

    therefore

    [tex]\lambda = \pm 3i[/tex]
    (complex roots)

    so
    [tex]y_{aux} = A cos 3x + B sin 3x[/tex]

    but now how do I get [tex]y_{particular}[/tex]??
     
  2. jcsd
  3. Dec 6, 2009 #2

    rock.freak667

    User Avatar
    Homework Helper

    Notice that '3' is a root in the auxiliary equation.

    what is the PI for sin3x? Since '3' is a root, you will need to modify the PI by multiplying it by x.
     
  4. Dec 6, 2009 #3

    Mark44

    Staff: Mentor

    PI = particular ???

    To expand on what rock.freak667 said, your particular solution should be yp = Axcos(3x) + Bxsin(3x).
     
  5. Dec 6, 2009 #4
    I don't really understand what you mean by this..

    However, the PI for sin3x is Acos3x + Bsin3x, i think...
    so if you have to multiply by x (don't really understand why?) then the PI would be
    x(Acos3x + Bsin3x)?
     
  6. Dec 6, 2009 #5
    PI = Particular Integral :)
     
  7. Dec 6, 2009 #6

    Mark44

    Staff: Mentor

    Not if the complementary solution yc is c1cos3x + c2sin3x, which means that no matter what the values of c1 and c2, yc'' + 9yc = 0. In other words, there is no way you will end up with 36sin3x on the right side.

    Your particular solution (I prefer this term to particular integral) must therefore be Axcos3x + Bxsin3x. You need to find the coefficients A and B so that yp'' + 9yp = 36sin(3x).

    Your general solution will be y = yc + yp = c1cos3x + c2sin3x + Axcos3x + Bxsin3x, where you will have determined A and B.
     
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