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Differential Equations help

  1. Feb 11, 2014 #1
    Hello all, I am currently having trouble with this Differential Equations problem.

    Let x = F(t) be the general solution of x'=P(t)x+g(t), and let x=V(t) be some particular solution of the same system. By considering the difference F(t)−V(t), show that F(t)=U(t)+V(t), where U(t) is the general solution of the homogeneous system x'=P(t)x.

    Attempt:

    Since F is a solution, we know that there exists a fundamental matrix such that M such that F=MW, where W is such that MW′=g. But that is all I have been able to deduce. Also, I am not sure if F(t) - V(t) would be considered a soluton as well.

    Thank you for your time. :)
     
    Last edited: Feb 11, 2014
  2. jcsd
  3. Feb 12, 2014 #2

    Simon Bridge

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    Welcome to PF;
    It looks like you are being asked to prove the usual theorem that is used to solve non-homogeneous DEs - generalized for a system of DEs. You can always look up how it is normally done for clues.

    Label your equations - (1) is the inhomogeneous equation and (2) is the associated homogeneous one.
    So F is the general solution to (1) and V is a particular solution to (1).
    You can easily check to see if F-V is a solution to (1) - plug it in.
     
    Last edited: Feb 12, 2014
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