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Math Help - Differential Equations and Variation of Parameters

  1. #16
    Senior Member tukeywilliams's Avatar
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    For the last solution,

    y'''+y'=sec t

    You can let z=y'

    To get,

    z''+z=sec t

    The general solution to homogenous equation is:
    z=C_1sin(t)+C_2cos(t)

    Then you can find particular via Lagrange's Variation of Parameters techiqnue.

    Then you need to integrate the entire solution for z tto get y.
    Couldnt you let L(y) = y''' + p(t)y'' + q(t)y' + r(t)y = f(t)

    and y_h = c_1y_! + c_2y_2 + c_3y_3 and y_p = v_1y_1+v_2y_2+v_3y_3

    Then v_1', v_2, and v'_3 have to satisfy some auxillary condition, and you can use Crameris Rule to get the solution.
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  2. #17
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    Quote Originally Posted by tukeywilliams View Post
    Couldnt you let L(y) = y''' + p(t)y'' + q(t)y' + r(t)y = f(t)

    and y_h = c_1y_! + c_2y_2 + c_3y_3 and y_p = v_1y_1+v_2y_2+v_3y_3

    Then v_1', v_2, and v'_3 have to satisfy some auxillary condition, and you can use Crameris Rule to get the solution.
    I am not sure what you are doing.

    There is a Generalized Variation of Parameters.

    But again it involves finding all 3 independent solution to,
    y'''+y'=0
    Which is,
    z''+z=0 (after reduction of order).
    Hence,
    z=y'=C_1sin(t)+C_2cos(t)
    But then,
    y=C_1sin(t)+C_2cos(t)+C_3

    Thus,
    y_1=sin(t)
    y_2=cos(t)
    y_3=1
    Are three linearly independent solutions.

    Now! You can use the generalized variation of parameters techinique for linear differencial equations of order 3.
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  3. #18
    Senior Member tukeywilliams's Avatar
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    Heres what I did roughly:

    Here
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  4. #19
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    Quote Originally Posted by tukeywilliams View Post
    Heres what I did roughly:

    Here
    It looks cool, but I have no idea what you did.
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  5. #20
    Senior Member tukeywilliams's Avatar
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    I edited it:

    Here
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  6. #21
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    Quote Originally Posted by tukeywilliams View Post
    I edited it:

    Here
    That looks right.
    W is the Wronskian (which you know because of my earlier post).

    I see what you are saying.
    That is the standard way I believe multi-dimensional variation of parameters is set up. However, that is not the solution. You need to painfully evaluate each determinant and them integrate all three v_1',v_2',v_3'.

    It takes a longgg time to do that.

    ----
    By the Way: You write like a girl!
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