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Math Help - Help with a Dif EQ

  1. #1
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    Help with a Dif EQ

    Consider the following differential equation:
    x''+x'+4x=ϵsin(2t)
    where ϵ is a free parameter. This system belongs to the class of damped driven harmonic oscillators.

    Find a particular solution that solves the inhomogeneous differential equation.
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  2. #2
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    Quote Originally Posted by splash View Post
    Consider the following differential equation:
    x''+x'+4x=ϵsin(2t)
    where ϵ is a free parameter. This system belongs to the class of damped driven harmonic oscillators.

    Find a particular solution that solves the inhomogeneous differential equation.
    It helps knowing the independent solutions.
    Just by looking at the quadradic it seems that e*sin(2t) is not among them.

    Thus, by undetermined coefficients we look for a solution of the form:
    x=A*sin(2t)+B*cos(2t)
    x'=2A*cos(2t)-2B*sin(2t)
    x''=-2A*sin(2t)-2B*cos(2t)

    Thus,
    [-2B-3A]sin(2t)+[2A-B]cos(2t)=e*sin(2t).

    Thus,
    -2B-3A=e
    -B+2A=0

    The solution to this system provides a particular solution.
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