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TT2 #2

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Tao Hu:
A typed solution may be more helpful :)
2(a):
\begin{equation*} \textbf{x}'=\begin{pmatrix}\hphantom{-}0 & 1\\\hphantom{-}2 &1 \end{pmatrix}\textbf{x}\ . \end{equation*}

find eigenvalues

\begin{equation*} r^2 - trace(A) + (ad- bc) =  r^2 -r - 2 = 0\implies r_1= 2, r_2=-1\end{equation*}

then, find eigenvectors, when r = 2

\begin{equation*} \begin{pmatrix} 0 - 2 & \hphantom{-}1\\  \hphantom{-}2 &1 -2\end{pmatrix}\begin{pmatrix}\mathbf{\xi}{^1}{_1}\\\mathbf{\xi}{^1}{_2}\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix} \end{equation*}

then

\begin{equation*}\mathbf{\xi}^1 =\begin{pmatrix}1\\2\end{pmatrix}\end{equation*}

when r = -1

\begin{equation*} \begin{pmatrix} 0  + 1 & \hphantom{-}1\\  \hphantom{-}2 &1 +1\end{pmatrix}\begin{pmatrix}\mathbf{\xi}{^2}{_1}\\\mathbf{\xi}{^2}{_2}\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix} \end{equation*}

then

\begin{equation*}\mathbf{\xi}^2=\begin{pmatrix}1\\-1\end{pmatrix}\end{equation*}

Therefore

\begin{equation*}\mathbf{x}(t)= C_1e^{2t}\begin{pmatrix}1\\2\end{pmatrix}+ C_2e^{-t}\begin{pmatrix}1\\-1\end{pmatrix}\end{equation*}

Real eigenvalues with distinct signs, the type of origin is a saddle point.

2(b):

\begin{equation*} \mathbf{x}(0)=C_1+ C_2=2\\\mathbf{y}(0)=2C_1- C_2=1 \end{equation*}
Easy Calculation:
\begin{equation*} C_1 = 1, C_2 = 1 \end{equation*}
final answer:
\begin{equation*}\mathbf{x}(t)= e^{2t}\begin{pmatrix}1\\2\end{pmatrix}+ e^{-t}\begin{pmatrix}1\\-1\end{pmatrix}\end{equation*}

Victor Ivrii:

--- Quote from: Tao Hu on November 20, 2014, 05:09:36 PM ---
--- End quote ---
Indeed

Yuan Bian:
I agree with you that typed version is better. As math students, I think many students wants to get bonus by answering online question and using typed version. but situation now, it's few of us can type math formula using latex(? or something else....),
As a math course, I think we should focus on math idea and correctness.
and I think copy my answer above to get bonus is not fair, and even not sure I get bonus or not.   

Victor Ivrii:
You got bonus. But bonus in this case is for the usefulness to community (for math part everyone gets much more as a regular mark for a test).  In this case typed solution which could be edited by the author or by a moderator is much more useful

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