# Toronto Math Forum

## MAT244-2018S => MAT244--Tests => Quiz-6 => Topic started by: Victor Ivrii on March 16, 2018, 08:08:07 PM

Title: Q6--T0101
Post by: Victor Ivrii on March 16, 2018, 08:08:07 PM
a. Express the general solution of the given system of equations in terms of real-valued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as $t\to \infty$.

$$\mathbf{x}' =\begin{pmatrix} 2 &-5\\ 1 &-2 \end{pmatrix}\mathbf{x}$$
Title: Re: Q6--T0101
Post by: Cheng Sheng on March 17, 2018, 01:51:27 AM
The steps are shown below.
Title: Re: Q6--T0101
Post by: Victor Ivrii on March 17, 2018, 05:11:01 AM

Why is your picture of that colour?!!
Title: Re: Q6--T0101
Post by: Junya Zhang on March 18, 2018, 11:37:36 AM
Cheng Sheng's solution is correct, but here's the typed solution :)
a)
Let $$P=\begin{pmatrix} 2 & -5 \\ 1 & -2\end{pmatrix}$$
Characteristic polynomial:
$$\chi_P(\lambda) = det\begin{pmatrix} 2-\lambda & -5 \\ 1 & -2-\lambda\end{pmatrix}=\lambda^2 + 1$$
Thus, $$\lambda_1 = i, \lambda_2 = -i$$
Consider $\lambda_1 = i$.
$$N(P-iI) = N\begin{pmatrix} 2-i & -5 \\ 1 & -2-i\end{pmatrix} = span\{\begin{pmatrix} 2+i \\ 1\end{pmatrix}\}$$
Consider $$e^{it}\begin{pmatrix} 2+i \\ 1\end{pmatrix} = \cos(t) + i\sin(t)\begin{pmatrix} 2+i \\ 1\end{pmatrix}\ = \begin{pmatrix} 2\cos(t) +2i\sin(t)+i\cos(t) - \sin(t) \\ \cos(t) + i\sin(t) \end{pmatrix} =\begin{pmatrix} 2\cos(t) - \sin(t) \\ \cos(t) \end{pmatrix} + i \begin{pmatrix} 2\sin(t) + \cos(t) \\ \sin(t)\end{pmatrix}$$
Thus, the general solution of this system is
$$\textbf{x}(t) = c_1\begin{pmatrix} 2\cos(t) - \sin(t) \\ \cos(t) \end{pmatrix} + c_2 \begin{pmatrix} 2\sin(t) + \cos(t) \\ \sin(t)\end{pmatrix}$$

b)
As $t\to\infty$, solution circulates in counter clockwise direction in elliptical shapes.
See attached image.
Title: Re: Q6--T0101
Post by: Victor Ivrii on March 18, 2018, 11:44:41 AM
\det  should be also "escaped" (\ is an escape character)