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Math Help - Justifying the Method of Undetermined Coefficients

  1. #1
    Senior Member ecMathGeek's Avatar
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    Justifying the Method of Undetermined Coefficients

    I'm trying to "Justify the Method of Undetermined Coefficients" by using the "Annihilator Method."
    To start, I am given a linear, constant coefficient, nonhomogeneous differential equation:

    L[y] = g(x)

    Where
    L[y] = s_{k}*y^{k} + s_{k-1}*y^{k-1} + ... + s_{0}y
    [Note: y^{k} means the k'th derivative of y.]

    And
    g(x) = p_{n}(x)*e^(ax)cos(bx) + q_{m}(x)*e^(ax)cos(bx)
    [Note: n and m are the degrees of the polynomials p and q, respectively.]

    ---------------------------------------------------------------------------------
    Among the things I need to do is:

    1. Show that:
    A := [(D-a)^2 + b^2]^(N+1)
    is an annihilator for g(x), where N := max(n,m)

    2. Show that the auxiliary equation associated with AL[y] = 0 is of the form:
    s_{k}*[(r - a)^2 + b^2]^(u+N+1)*(r - r_{2u+1})*...*(r - r_{k}) = 0
    where u >= 0 is the multiplicity of a +/- ib as roots of the auxiliary equation associated with
    L[y] = 0, and r_{2u+1},...,r_{n} are the remaining roots of this equation.

    ---------------------------------------------------------------------------------
    Here's the help I need:

    1. I don't know how to prove that A := [(D-a)^2 + b^2]^(N+1) is an annihilator for g(x). To
    "prove" this would take tons of work and dozens of pages worth of writing (I think). I'm not sure
    if I'm expected to "show" this is the annihilator by showing all the work or if I'm just expected to
    explain why it's true.

    2. I know the assumed form of the solutions to AL[y] = 0 is y = e^(rt), so to find the auxiliary
    equation I would need to find AL[e^(rt)], but I'm not sure how to plug this in.

    3. Also, if you notice, the auxiliary equation has [(r - a)^2 + b^2]^(u+N+1), where
    u is the multiplicity of the complex roots. I have no clue how to get that.

    Any help in trying to get through this will be greatly appreciated.
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  2. #2
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by ecMathGeek View Post
    2. I know the assumed form of the solutions to AL[y] = 0 is y = e^(rt), so to find the auxiliary
    equation I would need to find AL[e^(rt)], but I'm not sure how to plug this in.
    I figured out how to do this. I simply expanded the differential operator AL and showed that after plugging in e^(rt), the form of the auxiliary equatoin has the same form as that of the expanded operator, and thus when factored, the auxiliary equation will have the same form as the factored form of the operator.

    After figuring this out, I came across something else I cannot explain.

    Apparently I need to show how the factored from of the auxiliary equation will take the form:
    s_{k}*[(r - a)^2 + b^2]^(u + K + 1)*(r - r_{2u+1})***(r - r_{k}) = 0

    where r_{2u+1},...,r_{k} are the remaining roots of the auxiliary equation after factoring out the "u" complex roots. I have no clue where r_{2u+1} or r_{k} come from. I might eventually figure this out, but right now I'm stumped.
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  3. #3
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by ecMathGeek View Post
    I figured out how to do this. I simply expanded the differential operator AL and showed that after plugging in e^(rt), the form of the auxiliary equatoin has the same form as that of the expanded operator, and thus when factored, the auxiliary equation will have the same form as the factored form of the operator.

    After figuring this out, I came across something else I cannot explain.

    Apparently I need to show how the factored from of the auxiliary equation will take the form:
    s_{k}*[(r - a)^2 + b^2]^(u + K + 1)*(r - r_{2u+1})***(r - r_{k}) = 0

    where r_{2u+1},...,r_{k} are the remaining roots of the auxiliary equation after factoring out the "u" complex roots. I have no clue where r_{2u+1} or r_{k} come from. I might eventually figure this out, but right now I'm stumped.
    I'm starting to suspect no one will help me. It's fine if no one does. I'll just struggle through this problem and hopefully get it done. Anyways, I have figured out where these terms are coming from.

    Since the complex roots to the auxiliary equation of L are:
    [(r - a)^2 + b^2]^u

    Where there are "u" complex conjugate roots, there are actually a total of "2u" roots being taken care of by this term. Therefore, there are k - 2u remaining roots to this auxiliary equation, starting with r_{2u + 1}, r_{2u + 2},...,r_{k}.
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  4. #4
    Senior Member ecMathGeek's Avatar
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    The only question I still have is this.

    Quote Originally Posted by ecMathGeek View Post
    1. I don't know how to prove that A := [(D-a)^2 + b^2]^(N+1) is an annihilator for g(x). To
    "prove" this would take tons of work and dozens of pages worth of writing (I think). I'm not sure
    if I'm expected to "show" this is the annihilator by showing all the work or if I'm just expected to
    explain why it's true.
    I'm not sure if it's possible to prove that A is an annihilator for g(x) without dozens of pages of writing.

    Is there a quick way to show A works?
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