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2nd ODE

  1. Mar 17, 2006 #1
    Hi I have problem to answer this question. I think that it is a question of recognizing the right solution rather than computing.

    Give the general solution of the differential equation:
    y'' -6y'+9y=-5e(3x)+(1/x)e(3x)

    1- y=C1*e(3x)+C2*x*e(3x)-(5/2)x2*e(3x)+x*e(3x)*ln(x)

    2- y=C1*e(3x)+C2*x*e(3x)-(5/2)x*e(3x)-x*e(3x)*ln(x)


    My problem is when to know if the particular solution is multiply by x or x^2.
    when the particular solution is a solution of the homogenous equation , we multiply the particular solution by x. when do we multiply by x^2??
    Can you give me a quick example please??
    Thank you
  2. jcsd
  3. Mar 17, 2006 #2
    you would multiply the paticular solution by x^2 when there is a repeated root of multiplicity of 3.
    In other words if the solution had a triple root of 1.

    a*e^x + b*x*e^x + c*x^2*e^x
  4. Mar 18, 2006 #3


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    Staff Emeritus
    Science Advisor

    It should be obvious that the correct answer can't be 2 or 3 because
    -(5/2)*e(3x) could be absorbed into the C1*e(3x) term and (5/2)x*e(3x) could have been absorbed in to C2*x*e(3x). Those don't give anything new.
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