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Math Help - ODE Matrix exponential

  1. #1
    Junior Member
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    Nov 2008
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    Angry ODE Matrix exponential

    Identify the following using known results about the matrix exponential.

    e. Suppose the pair (x(t), y(t)) is a solution to an autonomous first order system whose matrix exponential is e^(tA).
    If (x(3),y(3)) = (5,7), how can (x(5),y(5)) be expressed?????????
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  2. #2
    Senior Member
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    Recall that the first-order linear system of ODE given by X'=AX, where X(t)=\begin{bmatrix}x(t)\\y(t)\end{bmatrix} is the (column) vector of our multivariable solution, has solution given by the matrix exponential X(t)=e^{tA}X_0, where X_0=\begin{bmatrix}x(0)\\y(0)\end{bmatrix} is our initial value. So we have in our problem that X(3)=e^{3A}X_0=\begin{bmatrix}5\\7\end{bmatrix}.
    Matrix exponentials have the property e^{sA}e^{tA}=e^{(s+t)A}, so
    X(5)=e^{5A}X_0
    \;=e^{2A}e^{3A}X_0
    \;=e^{2A}\begin{bmatrix}5\\7\end{bmatrix}.

    --Kevin C.
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