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Fourth Order Homogenous Differential Equation

  1. Nov 23, 2009 #1
    1. The problem statement, all variables and given/known data
    y4 - 6y3 + 9y2
    y(0) = 19; y'(0) = 16; y"(0) = 9; y"'(0) = 0

    2. Relevant equations

    3. The attempt at a solution
    Factored out the equation and obtained the following roots.
    r2(r2-3) = 0 which gives r = 0 and r =3.
    Using those roots, I make the following general solution.
    y(x) = j*e3x + k*x*e3x + L*k*x2*e3x
    I am assuming since one of the roots is zero then the solution will not have to have add i*e0t. I am also assuming since this is a fourth order equation that I will need to solve for n=4 variables and that this is the form in which I should tackle it. Am I mistaken in my assumptions?
  2. jcsd
  3. Nov 23, 2009 #2


    Staff: Mentor

    Is your differential equation? y(4) - 6y(3) + 9y(2) = 0?

    Your characteristic equation has four roots, with 0 and 3 repeated. Your basic set of solutions is {1, x, e3x, xe3x}. Your general solution will be all linear combinations of these functions, or
    y(x) = c1*1 + c2*x + c3*e3x + c4*xe3x. Use your initial conditions to solve for the constants ci.
  4. Nov 23, 2009 #3
    Yea it was equal to zero and thanks for the general equation. It solved a lot of the confusion I had on what to do with the r = 0.
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