12-267/Homework Assignment 6

This assignment is due in class on Friday November 9. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.

Task 1. Draw the phase portraits for the following systems, near $(x,y)=(0,0)$:

1. $\begin{cases} \dot{x}=2x+y \\ \dot{y}=-x+4y \end{cases}$.
2. $\begin{cases} \dot{x}=4x-5y \\ \dot{y}=4x-4y \end{cases}$.
3. $\begin{cases} \dot{x}=x-2y \\ \dot{y}=-2x+4y \end{cases}$.
4. $\begin{cases} \dot{x}=-x+y \\ \dot{y}=-5x+3y \end{cases}$.
5. $\begin{cases} \dot{x}=-5x+4y \\ \dot{y}=-8x+7y \end{cases}$.

Task 2. Draw the phase portrait of the system

$\begin{cases}\dot{x}=17+x-9y+\sin(2-2x-y+xy)\\\dot{y}=7+2x-5y+\cos(x-1)\end{cases}$

near the point $(x,y)=(1,2)$.

Task 3. Solve using diagonalization (one solution is enough):

1. $v'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} e^t \\ t \end{pmatrix}$.
2. $v'=\begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix}v + \begin{pmatrix} -\cos t \\ \sin t \end{pmatrix}$.

Task 4. Assume $t>0$. For the following equation,

$tv'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} 1-t^2 \\ 2t \end{pmatrix}$
,

it is given that a solution of the homogeneous version is

$v(t) = c_1\begin{pmatrix}1\\1\end{pmatrix}t + \begin{pmatrix}1\\3\end{pmatrix}t^{-1}$.

Use "fundamental solutions" to find a solution of the full equation.