Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations. displaystyle{leftlbracebegin{matrix}{x}+{y}={0}{5}{x}-{2}{y}-{2}{z}={12}{2}{x}+{4}{y}+{z}={5}end{matrix}right.}

Find the linear approximation of the function $f(x)=\sqrt{4-x}$ at $a=0$
Use L(x) to approximate the numbers $\sqrt{3.9}$ and $\sqrt{3.99}$ Round to four decimal places

Write the following linear differential equations with constant coefficients in the form of the linear system ${\displaystyle \dot{x}=Ax}$ and solve:
${\displaystyle \left(a\right)\ddot{x}+\dot{x}-2x=0}$
${\displaystyle \left(b\right)\ddot{x}+x=0}$
${\displaystyle \left(c\right)d\ddot{x}-2\ddot{x}-\dot{x}+2x=0}$
Hint: Let ${\displaystyle x_1=x,x_2=\dot{x}}$, etc.
I have tried to do this in the following way but I do not know if I am doing well:
Let ${\displaystyle x_1=x,x_2=\dot{x},x_3=\ddot{x},x_4=x^{\dots }}$, then the previous system becomes ${\displaystyle \left(a\right)x_3+x_2-2x_1=0,\left(b\right)x_3+x_1=0,\left(c\right)x_4-x_3-x_2+2x_1=0}$
and thus ${\displaystyle \left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{c}\frac{x_2}{3}\\ x_2\\ \frac{-x_2}{3}\\ 0\end{array}\right]}$. Is this what it should be or is it different? Thank you very much.

Consider the following linear system of differential equations:
$\{\begin{array}{l}\dot{x}=-4y\\ \dot{y}=x\end{array}$
where $x(t)$ and $y(t)$ are unknown real functions.
One can simply verify that the general solution is
$\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)=c_1\left(\begin{array}{c}\cos (2t)\\ \frac{1}{2}\sin (2t)\end{array}\right)+c_2\left(\begin{array}{c}-2\sin (2t)\\ \cos (2t)\end{array}\right)$
where $c_1$ and $c_2$ are real parameters.
Question: Which is the exponential form of this expression? It should by something like
$\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)=k_1{e}^{i2t}\left(\begin{array}{c}{P}_{11}\\ P_{21}\end{array}\right)+k_2{e}^{-i2t}\left(\begin{array}{c}{P}_{12}\\ P_{22}\end{array}\right)$
where $k_1$ and $k_2$ should (?) be complex parameters.

Obtain just a particular solution (the general solution can alwaysbe obtained easily by adding an arbitrary multiple of a solution of the associated homogeneous equation). In these exercises, the forcing function is an elementary sinusoidal function. If the forcing function is ${\displaystyle \cos 5t}$, then a particular solution of the form A ${\displaystyle \cos 5t}$ will not work. However, a linear combination of ${\displaystyle \cos 5t}$ and ${\displaystyle \sin 5t}$ will work. Thus, the form for xp is ${\displaystyle A\cos 5t+B\sin 5t}$, where the constants A and B are determined by substituting the assumed form of the particular solution into the differential equation for x. In Exercises 47–52, the forcing function has several different terms. Use a form for xp consisting of the sum of the forms for each term.
${\displaystyle \frac{dx}{dt}+x=2e^{3t}+\sin t}$

I mean, that whenever we talk about linear equations, we say that they always form a straight line on a graph. We make this sure by making a graph of some possible solutions to the equation and plotting the points on the graph. But can we make sure that all the solutions of the linear equation will always form a straight line because we cannot plot each and every solution of the equation on the graph because there are infinitely many solutions to a linear equation in 2 variables?

How to solve the system of linear differential equation of the form
${\displaystyle x^{\prime }=Ax+b}$
I can solve the homogeneous form by finding the eigenvalues and respective eigenvectors, but how to find the particular solution part. Also is there any limitation from getting eigenvalues positive, negative or complex. Any other different method involving matrix algebra is also welcome.

The linear equation is the equation that has a from:
${\displaystyle a_1{x}_1+a_2{x}_2+\dots +a_n{x}_n=b}$
But is this really the definition of a linear equation? I thought that the definition should be the form of linear mapping. Like,
${\displaystyle f(x+y)=f\left(x\right)+f\left(y\right)}$
${\displaystyle f\left(ax\right)=af\left(x\right)}$
What is the definition of a linear equation?