It looks like this is a system of linear ODEs given in matrix form,
![x' = \begin{bmatrix}10&-1\\5&8\end{bmatrix} x](https://tex.z-dn.net/?f=x%27%20%3D%20%5Cbegin%7Bbmatrix%7D10%26-1%5C%5C5%268%5Cend%7Bbmatrix%7D%20x)
with initial condition x(0) = (-6, 8)ᵀ.
Compute the eigenvalues and -vectors of the coefficient matrix:
![\det\begin{bmatrix}10-\lambda&-1\\5&8-\lambda\end{bmatrix} = (10-\lambda)(8-\lambda) + 5 = 0 \implies \lambda^2-18\lambda+85=0 \implies \lambda = 9\pm2i](https://tex.z-dn.net/?f=%5Cdet%5Cbegin%7Bbmatrix%7D10-%5Clambda%26-1%5C%5C5%268-%5Clambda%5Cend%7Bbmatrix%7D%20%3D%20%2810-%5Clambda%29%288-%5Clambda%29%20%2B%205%20%3D%200%20%5Cimplies%20%5Clambda%5E2-18%5Clambda%2B85%3D0%20%5Cimplies%20%5Clambda%20%3D%209%5Cpm2i)
Let v be the eigenvector corresponding to λ = 9 + 2i. Then
![\begin{bmatrix}10-\lambda&-1\\5&8-\lambda\end{bmatrix}v = 0 \implies \begin{bmatrix}1-2i&-1\\5&-1-2i\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D10-%5Clambda%26-1%5C%5C5%268-%5Clambda%5Cend%7Bbmatrix%7Dv%20%3D%200%20%5Cimplies%20%5Cbegin%7Bbmatrix%7D1-2i%26-1%5C%5C5%26-1-2i%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7Dv_1%5C%5Cv_2%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D0%5C%5C0%5Cend%7Bbmatrix%7D)
or equivalently,
![\begin{cases}(1-2i)v_1-v_2=0 \\ 5v_1-(1+2i)v_2=0\end{cases} \implies 5v_1 - (1+2i)v_2 = 0](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%281-2i%29v_1-v_2%3D0%20%5C%5C%205v_1-%281%2B2i%29v_2%3D0%5Cend%7Bcases%7D%20%5Cimplies%205v_1%20-%20%281%2B2i%29v_2%20%3D%200)
Let
; then
, so that
![\begin{bmatrix}10&-1\\5&8\end{bmatrix}\begin{bmatrix}1\\1-2i\end{bmatrix} = (9+2i)\begin{bmatrix}1\\1-2i\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D10%26-1%5C%5C5%268%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7D1%5C%5C1-2i%5Cend%7Bbmatrix%7D%20%3D%20%289%2B2i%29%5Cbegin%7Bbmatrix%7D1%5C%5C1-2i%5Cend%7Bbmatrix%7D)
and we get the other eigenvalue/-vector pair by taking the complex conjugate,
![\begin{bmatrix}10&-1\\5&8\end{bmatrix}\begin{bmatrix}1\\1+2i\end{bmatrix} = (9-2i)\begin{bmatrix}1\\1+2i\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D10%26-1%5C%5C5%268%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7D1%5C%5C1%2B2i%5Cend%7Bbmatrix%7D%20%3D%20%289-2i%29%5Cbegin%7Bbmatrix%7D1%5C%5C1%2B2i%5Cend%7Bbmatrix%7D)
Then the characteristic solution to the system is
![x = C_1 e^{(9+2i)t} \begin{bmatrix}1\\1-2i\end{bmatrix} + C_2 e^{(9-2i)t} \begin{bmatrix}1\\1+2i\end{bmatrix}](https://tex.z-dn.net/?f=x%20%3D%20C_1%20e%5E%7B%289%2B2i%29t%7D%20%5Cbegin%7Bbmatrix%7D1%5C%5C1-2i%5Cend%7Bbmatrix%7D%20%2B%20C_2%20e%5E%7B%289-2i%29t%7D%20%5Cbegin%7Bbmatrix%7D1%5C%5C1%2B2i%5Cend%7Bbmatrix%7D)
From the given condition, we have
![\displaystyle \begin{bmatrix}-6\\8\end{bmatrix} = C_1 \begin{bmatrix}1\\1-2i\end{bmatrix} + C_2 \begin{bmatrix}1\\1+2i\end{bmatrix} \implies C_1 = -3-\frac i2, C_2=-3+\frac i2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Bbmatrix%7D-6%5C%5C8%5Cend%7Bbmatrix%7D%20%3D%20C_1%20%5Cbegin%7Bbmatrix%7D1%5C%5C1-2i%5Cend%7Bbmatrix%7D%20%2B%20C_2%20%5Cbegin%7Bbmatrix%7D1%5C%5C1%2B2i%5Cend%7Bbmatrix%7D%20%5Cimplies%20C_1%20%3D%20-3-%5Cfrac%20i2%2C%20C_2%3D-3%2B%5Cfrac%20i2)
and so the particular solution to the IVP is
![\displaystyle \boxed{x = -\left(3+\frac i2\right) e^{(9+2i)t} \begin{bmatrix}1\\1-2i\end{bmatrix} - \left(3-\frac i2\right) e^{(9-2i)t} \begin{bmatrix}1\\1+2i\end{bmatrix}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cboxed%7Bx%20%3D%20-%5Cleft%283%2B%5Cfrac%20i2%5Cright%29%20e%5E%7B%289%2B2i%29t%7D%20%5Cbegin%7Bbmatrix%7D1%5C%5C1-2i%5Cend%7Bbmatrix%7D%20-%20%5Cleft%283-%5Cfrac%20i2%5Cright%29%20e%5E%7B%289-2i%29t%7D%20%5Cbegin%7Bbmatrix%7D1%5C%5C1%2B2i%5Cend%7Bbmatrix%7D%7D)
which you could go on to rewrite using Euler's formula,
![e^{(a+bi)t} = e^{at} (\cos(bt) + i \sin(bt))](https://tex.z-dn.net/?f=e%5E%7B%28a%2Bbi%29t%7D%20%3D%20e%5E%7Bat%7D%20%28%5Ccos%28bt%29%20%2B%20i%20%5Csin%28bt%29%29)
The way elements are used in the different genres is that Genre as it s made up 4 elements such as character, story, plot and setting, they are said to be mixed up together.
They can be mixed up as Story (Action) + Plot + Character + Setting = Genre. Some other kinds of genres are crime, war, spy, adventure, etc.
Note that a lot of genres or forms of literature often depicts themes or topics in a different way.
To Compare and Contrast genre, one must shows what is similar and what is different about the genre in question. Knowing how the genres are alike and difference can helps a person to have a better study of genre.
<h3>What is the differences in the use of genre/</h3>
A lot of sub-genres of comedy are known to be different from one another when one looks at the fluctuations in terms of the characters and the story.
The Characteristics of Genre are:
- Form: This shows one how a writer makes, structures, and put together a written work.
- Style: This is known to be a method used by writer in terms of words and language to pass out meaning, mood, etc.
- Subject: This is known to be the main topic the genre is based on, etc.
Learn more about genres from
brainly.com/question/1309721
Answer: <em><u>7.38293375E17</u></em>
The best option in the set of questions being asked in an Aspire test would be c. Multiple choice, short answer questions and fill in the blank questions. The Response tasks would require students to demonstrate their knowledge by selecting the appropriate response.
I need the whole question