Initial Value Problem
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Exercise:
A thredimensional system of differential s is characterised by the eigenvalues lambda_ - lambda_ lambda_ - and the corresponding eigenvectors vec v_ pmatrix pmatrix vec v_ pmatrix pmatrix vec v_ pmatrix pmatrix Derive the solution for the initial conditions pmatrix x y z pmatrix pmatrix - pmatrix
Solution:
The general solution is given by the vector pmatrix xt yt zt pmatrix a_ vec v_ e^lambda_ t + a_ vec v_ e^lambda_ t + a_ vec v_ e^lambda_ t For t this can be written as pmatrix x y z pmatrix pmatrix - pmatrix pmatrix a_ + a_ a_ a_+a_+a_ pmatrix From the second component it follows immediately that a_ The first component then yields a_ -a_ - - and the last component a_ - - a_ - a_ - + - - The solution can therfore be written as xt -e^-t+e^t yt e^t zt -e^-t+e^t-e^-t
A thredimensional system of differential s is characterised by the eigenvalues lambda_ - lambda_ lambda_ - and the corresponding eigenvectors vec v_ pmatrix pmatrix vec v_ pmatrix pmatrix vec v_ pmatrix pmatrix Derive the solution for the initial conditions pmatrix x y z pmatrix pmatrix - pmatrix
Solution:
The general solution is given by the vector pmatrix xt yt zt pmatrix a_ vec v_ e^lambda_ t + a_ vec v_ e^lambda_ t + a_ vec v_ e^lambda_ t For t this can be written as pmatrix x y z pmatrix pmatrix - pmatrix pmatrix a_ + a_ a_ a_+a_+a_ pmatrix From the second component it follows immediately that a_ The first component then yields a_ -a_ - - and the last component a_ - - a_ - a_ - + - - The solution can therfore be written as xt -e^-t+e^t yt e^t zt -e^-t+e^t-e^-t