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Abstract algebra: systems of differential linear equations

  1. Dec 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Solve the inhomogeneous differential equation dX/dt=AX+B in terms of the solutions to the homogeneous equation dX/dt=AX.


    2. Relevant equations

    A is an nxn real or complex matrix and X(t) is an n-dimensional vector-valued function.
    If v is an eigenvector for A with eigenvalue a, then X=v*ea*t is a particular solution to the differential equation dX/dt=AX.
    And the general solution of the homogenous eqn is X=P-1*Xtilda.


    3. The attempt at a solution

    So, the general solution of the inhomogeneous equation should be a particular solution of the inhomogenous equations + the general solution of the homogeneous equation. We know the general part, but I am lost on how to find the particular solution for an inhomogeneous equation. Any help would be apprecaited!
     
    Last edited: Dec 7, 2009
  2. jcsd
  3. Dec 7, 2009 #2

    CompuChip

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    Is B a constant vector? In that case, you could try a solution where X is also a constant vector, such that dX/dt = 0.
     
  4. Dec 7, 2009 #3

    HallsofIvy

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    You can use a multi-dimensional version of "variation of parameters". If the general solution to the homogeneous equation is [itex]X= P^{-1}X^~[/itex] try a solution of the from [itex]Y= P^{-1}X^~u(t)[/itex] where u(t) is an unknown function. Then [itex]Y'= P^{-1}X^~'u+ P^{-1}X^~u'[/itex] and [itex]Ay= AP^{-1}X^~u[/itex] so the equation becomes [itex]P^{-1}X^~'u+ P^{-1}X^~u'+ AP^{-1}X^~u= B[/itex].

    Since [itex]X= P^{-1}X^~[/itex] is a solution to the homogeneous equation, [itex]P^{-1}X^~'u+ AP^{-1}X^~u= (P^{-1}X^~'+ AP^{-1}X^~)u= 0[/itex] and the equation reduces to [itex]P^{-1}X^~u'= B[/itex] so [itex]u'= X^~^{-1}PB[/itex]. Integrate that to find u.
     
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