Results 1 to 5 of 5

Math Help - Applications of eigenvectors word problem

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    297

    Applications of eigenvectors word problem

    A radioactive isotope A decays at the rate of 2% per century into a second radioactive isotope B, which in turn decays at a rate of 1% per century into a stable isotope C.

    a) Find a system of linear differential equations to describe the decay process. If we start with pure A, what are the proportions of A, B and C after 500 years, after 1,000 years and after 1,000,000 years?

    This is what I've done so far:

    x_A(k)=\textrm{proportion of A after k centuries}
    x_B(k)=\textrm{proportion of B after k centuries}
    x_C(k)=\textrm{proportion of C after k centuries}

    x_A(k+1)=0.98x_A(k)

    x_B(k+1)=0.02x_A(k)+0.99x_B(k)

    x_C(k+1)=0.01x_B(k)+x_C(k)


    \mathbf{x}(k+1)=\begin{pmatrix}0.98&0&0\\0.02&0.99  &0\\0&0.01&1\end{pmatrix}\mathbf{x}(k)

    Let A=\begin{pmatrix}0.98&0&0\\0.02&0.99&0\\0&0.01&1\e  nd{pmatrix}

    \rightarrow \begin{pmatrix}0.98&0&0\\0&0.99&0\\0&0&1\end{pmatr  ix}

    I'm not too sure if I'm correct, I think the eigenvalues are 0.98, 0.99, 1

    At 500 years:

    \mathbf{x}(5)=A^5 \mathbf{x}(0)

    Is what I'm doing correct so far?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Differential equations? Or difference equations? Your approach is rather like a Markov chain; it could well be a valid approach, but I don't see any derivatives in there. In any case, how did you get this line:

    \rightarrow \begin{pmatrix}0.98&0&0\\0&0.99&0\\0&0&1\end{pmatr  ix}?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Feb 2008
    Posts
    297
    Quote Originally Posted by Ackbeet View Post
    Differential equations? Or difference equations? Your approach is rather like a Markov chain; it could well be a valid approach, but I don't see any derivatives in there. In any case, how did you get this line:

    \rightarrow \begin{pmatrix}0.98&0&0\\0&0.99&0\\0&0&1\end{pmatr  ix}?
    Just with row reduction
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Feb 2008
    Posts
    297
    Is this what you're talking about. How would I find the corresponding eigenvectors. So far I'm only used to finding eigenvectors of 2x2 matrices

    \mathbf{y}(t)=\alpha e^{0.98t}\begin{pmatrix}?\\?\\?\end{pmatrix}+\beta e^{0.99t}\begin{pmatrix}?\\?\\?\end{pmatrix}+\gamm  a e^{t}\begin{pmatrix}?\\?\\?\end{pmatrix}
    Follow Math Help Forum on Facebook and Google+

  5. #5
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    You can't use row reduction to diagonalize a matrix. So your candidate solution is, I'm afraid, incorrect. First, you need a differential equation model of the isotopes. What do you think is the correct model here?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: November 30th 2011, 09:29 PM
  2. Problem on Eigenvalues and Eigenvectors
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 14th 2011, 06:53 PM
  3. Linear Algebra, eigenvectors problem
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 15th 2010, 07:55 AM
  4. Algebra Applications/Word problems
    Posted in the Algebra Forum
    Replies: 3
    Last Post: December 21st 2009, 06:57 PM
  5. Replies: 4
    Last Post: September 15th 2009, 03:18 PM

Search Tags


/mathhelpforum @mathhelpforum