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Math Help - Initial-value problem using the method of eigenvalues/eigenvectors (Answer included).

  1. #1
    s3a
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    Initial-value problem using the method of eigenvalues/eigenvectors (Answer included).

    The biggest challenge for me in attempting this problem is using the Linear algebra.

    If I'm correct, I must:

    1) Find eigenvalues.
    2) Find eigenvectors.
    3) Compute e^(At)
    4) Multiply e^(At) by y_0 = y(0)

    So, assuming what I said is correct, it seems like I know what I am doing but it's in setting up each step and then taking the result and moving to the next step that loses me so I would really appreciate if someone could show me how to set up each step and the answer each step gives to feed into the next step. I don't need mechanical computations (software outputs should most likely be fine).

    Any help would be GREATLY appreciated!
    Thanks in advance!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Initial-value problem using the method of eigenvalues/eigenvectors (Answer includ

    Quote Originally Posted by s3a View Post
    If I'm correct, I must:
    1) Find eigenvalues.
    2) Find eigenvectors.
    3) Compute e^(At)
    4) Multiply e^(At) by y_0 = y(0)
    That is right, so let us go with 1) and 2). What eigenvalues and eigenvectors did you obtain?
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  3. #3
    s3a
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    Re: Initial-value problem using the method of eigenvalues/eigenvectors (Answer includ

    Sorry for the late reply, I was having difficulty computing things. I'm attaching my work so far. Hopefully, it's correct.

    Also, I know this is minor in comparison to my other problems but, instead of u_1, u_2, u_3 and w_1, w_2, w_3, should I have reused the v_i notation (strictly speaking) or was I technically correct to use different variables?
    Attached Thumbnails Attached Thumbnails Initial-value problem using the method of eigenvalues/eigenvectors (Answer included).-q9_mywork.jpg  
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Initial-value problem using the method of eigenvalues/eigenvectors (Answer includ

    Quote Originally Posted by s3a View Post
    Also, I know this is minor in comparison to my other problems but, instead of u_1, u_2, u_3 and w_1, w_2, w_3, should I have reused the v_i notation (strictly speaking) or was I technically correct to use different variables?
    It is irrelevant in our case. You also can use (for example) the notation x_1,x_2,x_3 for every eigenspace. Now, if P is the matrix whose columns are the eigenvalues associated to 4,-1,1 respectively, the solution is \vec{x}(t)=e^{tA}\vec{x}(0)=P\;\textrm{diag}\;(e^{  4t},e^{-t},e^t)\;P^{-1}\vec{x}(0)=\ldots
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