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Nonhomogeneous system particular solution.

  1. Apr 28, 2013 #1
    1. The problem statement, all variables and given/known data

    Verify that the vector functions [itex]x_{1}=\begin{bmatrix}e^{t}\\ e^{t}\end{bmatrix}[/itex] and [itex]x_{2}=\begin{bmatrix}e^{-t}\\ 3e^{-t}\end{bmatrix}[/itex] are solutions to the homogeneous system

    [itex]x'=Ax=\begin{bmatrix}2 & -1 \\ 3 & -2 \end{bmatrix}[/itex] on [itex](-\infty ,\infty )[/itex]

    and that

    [itex]x_{p} = \frac{3}{2}\begin{bmatrix}te^{t}\\ te^{t}\end{bmatrix}-\frac{1}{4}\begin{bmatrix}e^{t}\\3e^{t}\end{bmatrix} + \begin{bmatrix}t\\2t\end{bmatrix}-\begin{bmatrix}0\\1\end{bmatrix}[/itex] is a particular solution to the nonhomogeneous system x'=Ax + f(t), where f(t)=col([itex]e^{t},t[/itex]).

    Find a general solution to x' = Ax + f(t).

    2. Relevant equations

    I can handle the first part, basically showing with a Wronskian wether the solutions form a fundamental solution set.

    The second part, about the particular solution - completely stuck. I understand that the general solution is written in the form x = xp +Xc, so that is no problem.

    3. The attempt at a solution

    Should I use eigenvectors, or should consider another way, such as described here?

    http://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousSystems.aspx

    Really thankful for all the hints here because the chapter in the book does not even cover particular solutions.

    So, how do I verify that that particular solution is correct (meaning how do I derive a particular solution from the above data)?
     
  2. jcsd
  3. Apr 28, 2013 #2

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    IMO, all that is required is that you verify (i.e., plug and chug) that xp satisfies the inhomogeneous system.
    If it does, then write the general solution using xp and xc. I don't think you are expected to derive xp from scratch.
     
  4. Apr 29, 2013 #3

    Mark44

    Staff: Mentor

    Also, verify that x1(t) and x2(t) are solutions of the homogeneous system.
    Right. Problems like this are relatively simple. You're given a potential solution and are asked to confirm that it is indeed a solution. Substitute the given functions into the appropriate differential equation and you should end up with a statement that is identically true.
     
  5. Apr 29, 2013 #4
    Thanks for the advice but I'm sorry - this is not enough for me to understand. I need to find out the way to derive it, because I am certain that I will get this on the coming test.

    Here's what I found out on the them internets - but I got stuck here as well. I understand the general way to solving it, but maybe it's just that I'm stuck in some constants. Looking at the particular solution, this can be defines as:

    [itex]x_{p}=ate^{t}+be^{t}+ct+d[/itex]

    Looking at the formulas above, this can now be rewritten as:

    [itex]ate^{t}+(a+b)e^{t}+c=Aate^{t}+Abe^{t}+Act+Ad+ \begin{bmatrix}1 \\ 0 \end{bmatrix}e^{t}+ \begin{bmatrix} 0 \\ 1 \end{bmatrix}t[/itex]

    So now I have: [itex]Aa=a, Ab=a+b-\begin{bmatrix}1\\0\end{bmatrix},Ac=-\begin{bmatrix}0\\1\end{bmatrix}, Ad=c[/itex]

    Where A all along is [itex]A=\begin{bmatrix}2 & -1\\ 3 & -2\end{bmatrix}[/itex]

    I get the constant [itex]a=\frac{3}{2}[/itex] which is correct. However, I am completely stuck with the constant b. I am calculating using [itex]\begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}[/itex]. But I get two exactly same equations.

    Thank you for your help on this one!
     
    Last edited by a moderator: May 1, 2013
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