# Toronto Math Forum

## MAT244-2014F => MAT244 Math--Tests => TT2 => Topic started by: Victor Ivrii on November 19, 2014, 08:46:44 PM

Title: TT2 # 3
Post by: Victor Ivrii on November 19, 2014, 08:46:44 PM
Find the general solution, sketch the phase portrait and determine the type of behavior near the origin of the the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'_t= -x - 4y\ , \\
&y'_t=\ \ x -\ y .
\end{aligned}\right.
\end{equation*}
Title: Re: TT2 # 3
Post by: Yuan Bian on November 19, 2014, 10:33:18 PM
xâ€²=(âˆ’1 âˆ’4)(x) .
y'=(1    -1) (y)

find eigenvalues

det(Aâˆ’rI)=r2+2r+5=0
r1=-1+2i
r2=-1-2i
then, find eigenvectors

(-2i -4)   v1=(2i)
(1  -2i)         (1)

(2i -4)   v2=(2i)
(1  2i)         (-1)
x(t)=C1e(âˆ’1+2i)t(2i)+C2eâˆ’(-1-2i)t(2i)
(1)                      (-1)

stable spiral point
-4<0, counterclockwise
Title: Re: TT2 # 3
Post by: Chang Peng (Eddie) Liu on November 19, 2014, 10:37:09 PM
#3
Title: Re: TT2 # 3
Post by: Chang Peng (Eddie) Liu on November 19, 2014, 10:37:31 PM
Picture
Title: Re: TT2 # 3
Post by: Tao Hu on November 19, 2014, 10:38:32 PM
first write the equation in matrix form:

\begin{equation*}\textbf{x}'=\begin{pmatrix}\hphantom{-}-1 & -4\\\hphantom{-}1 &-1\end{pmatrix}\textbf{x}\ . \end{equation*}

find eigenvalues:

\begin{equation*} r^2 - trace(A) + (ad - bc)=  r^2+ 2r + 5 = 0\implies r_1= -1 + 2i,   r_2=-1 -2i\end{equation*}

then, find eigenvectors, which are conjugated

\begin{equation*} \begin{pmatrix} -1 - r & \hphantom{-}-4\\  \hphantom{-}1 &-1 -r\end{pmatrix}\begin{pmatrix}\mathbf{\xi}_1\\\mathbf{\xi}_2\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix} \end{equation*}

find the two conjugate eigenvectors

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

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

therefore

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

the attachment is the phase portrait generated by PPLANE
(spiral point, stable)
Title: Re: TT2 # 3
Post by: Victor Ivrii on November 20, 2014, 04:03:01 AM
Despite some glitches (one should write \sin (t) resulting in upright $\sin$ rather than sin (t) in LaTeX) the last solution is the only one I can read