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Thread: 27.1 Initial value problem

  1. #1
    Super Member bigwave's Avatar
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    27.1 Initial value problem

    27.1   Initial value problem-27.1b.png
    did a c/p just avoid typos

    ok the example did this
    $$A=\begin{pmatrix}1&2\\3&2 \end{pmatrix}$$
    and

    $$y'=Ay$$

    with initial value $$y(0)=\begin{pmatrix}y_1(0)\\y_2(0) \end{pmatrix}$$


    ok I'm ????
    Last edited by bigwave; Apr 4th 2019 at 06:13 PM.
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  2. #2
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    Re: 27.1 Initial value problem

    $y = c_1 e^{\lambda_1 t}v_1 + c_2 e^{\lambda_2 t} v_2$

    where $\lambda_k, v_k$ are the $kth$ eigenvalues/vectors of $A$

    Then use the initial conditions to solve for $c_1, c_2$
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  3. #3
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    Re: 27.1 Initial value problem

    $A=\begin{pmatrix}1 &2\\3 &2\end{pmatrix}$

    $y(0)=\begin{pmatrix}1\\3\end{pmatrix}$

    Find the eigensystem.

    $\left|\begin{pmatrix}1-\lambda &2\\3 &2-\lambda\end{pmatrix}\right| = \lambda^2-3\lambda -4$

    $\lambda^2-3\lambda-4 = 0\\
    (\lambda-4)(\lambda+1) = 0\\
    \lambda = 4,~-1
    $

    $\lambda_1=4 \Rightarrow v_1 = \begin{pmatrix}2\\3\end{pmatrix}$
    $\lambda_2=-1 \Rightarrow v_2 = \begin{pmatrix}1\\-1\end{pmatrix}$

    $y(t) = c_1 e^{4t}\begin{pmatrix}2\\3\end{pmatrix} + c_2 e^{-t}\begin{pmatrix}1\\-1\end{pmatrix}$

    $y(0) = c_1 \begin{pmatrix}2\\3\end{pmatrix} + c_2 \begin{pmatrix}1\\-1\end{pmatrix} = \begin{pmatrix}1\\3\end{pmatrix}$

    $(c_1,c_2) = \left(\dfrac 4 5, -\dfrac 3 5\right)$

    $y(t) = \dfrac 4 5 e^{4t}\begin{pmatrix}2\\3\end{pmatrix} - \dfrac 3 5 e^{-t}\begin{pmatrix}1\\-1\end{pmatrix}$


    $y(t) = \dfrac 1 5 \begin{pmatrix}8 e^{4t}-3 e^{-t}\\12 e^{4t} +3 e^{-t}\end{pmatrix}$
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  4. #4
    Super Member bigwave's Avatar
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    Re: 27.1 Initial value problem

    Appreciate much
    Have deal more with this tomorro
    To hard with just cell phone.
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  5. #5
    Super Member bigwave's Avatar
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    Re: 27.1 Initial value problem

    how did you get $\displaystyle v_1$ and $\displaystyle v_2$
    Last edited by bigwave; Apr 5th 2019 at 11:58 AM. Reason: tex
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  6. #6
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    Re: 27.1 Initial value problem

    Quote Originally Posted by bigwave View Post
    how did you get $\displaystyle v_1$ and $\displaystyle v_2$
    If you've advanced to the point of solving systems of linear diff eqs via eigensystems
    then you should know how to find eigenvectors given eigenvalues.

    Review your notes.
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