Calculus 2 Equations: Expert Solutions and More

Question on a differential equation
Let ${\displaystyle x:\mathbb{R}\Rightarrow \mathbb{R}}$ a solution of the differenzial equation:
${\displaystyle 5x{ }^{″}\left(t\right)+10x^{\prime }\left(t\right)+6x\left(t\right)=0}$
Proof that the function:
$f:\mathrm{ℝ}\Rightarrow \mathrm{ℝ}$

Limit for $\to \mathrm{\infty }$ of a solution to an ODE
Consider the Cauchy problem:
$\{\begin{array}{l}{y}^{\prime }(t)=\frac{t^2}{2+\sin (t^2)}cos(y(t{)}^2)\\ y(0)=0\end{array}$

How to solve this diffrential equation using parts formula?
$\frac{dy}{dx}+\frac{y}{20}=50(1+\cos x)$

Let ${\displaystyle f\in C_o^{\mathrm{\infty }}\left({\mathbb{R}}^n\right)}$. Propose formulas of the form $u\left(x\right)={\int }_{ }^{ }\frac{\hat{f}\left(\delta \right)e^{ix\delta }}{p\left(\delta \right)}d\delta$
to solve:
i) ${\displaystyle \sum _{j=1}^n\frac{{\partial }^{4}u\left(x\right)}{\partial x_j^4}+u\left(x\right)=f\left(x\right)}$
and

ii) $\sum _{k,l}^{ }\frac{{\partial }^{4}u\left(x\right)}{\partial x_k^2\partial x_l^2}-2·\sum _{k=1}^n\frac{{\partial }^{2}u\left(x\right)}{\partial x_k^2}+u\left(x\right)=f\left(x\right)$

Solve $x^{3}y\text{'}\text{'}\text{'}+xy\text{'}-y=x\ln \left(x\right)$
using shift ${\displaystyle x=e^z}$ and differential operator ${\displaystyle Dz=\frac{d}{ dz }}$.
What does ${\displaystyle Dz=\frac{d}{ dz }}$ mean?
$(e^z{)}^{3}y\text{'}\text{'}\text{'}+e^{z}y\text{'}-y=e^z\ln (e^z)$

$\left(e^{3z}\right)y\text{'}\text{'}\text{'}+e^{z}y\text{'}-y=e^{z}z$
${\displaystyle y=z^r}$
${\displaystyle e^{3}r(r^2-r-2)z^{r-3}+e^{z}r{z}^{r-1}-z^r=0}$
Continue ?

How can we show that $x\left(t\right)-y\left(t\right)={\int }_{s-\epsilon }^{t}f(x\left(r\right),a)-f(y\left(r\right),a\left(r\right)):\mathrm{dr}$ implies ${\displaystyle x\left(s\right)-y\left(s\right)=(f(x\left(s\right),a)-f(x\left(s\right),\alpha \left(s\right)))\epsilon +o\left(\epsilon \right)}$

How can I handle exp(ln|x|) to solve 1st order linear DE?
RHS and LHS are same.
$\exp (\ln \left(x\right))=\exp (\ln \left(x\right)) \left(1\right)$
Taking log.
$\ln (\exp (\ln \left(x\right)))=\ln (\exp (\ln \left(x\right))) \left(2\right)$
$\ln \left(x\right)=\ln (\exp (\ln \left(x\right))) \left(3\right)$
$\therefore \approx x=\exp (\ln \left(x\right)) \left(4\right)$
Then what about ln|x| ?
This problem is a sub-problem of below ODE.
$\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{2}{x}y=x^6 \left(5\right)$
My thoughts are as below.
$P\left(x\right)=-\frac{2}{x} \left(6\right)$
$Q\left(x\right)=x^6 \left(7\right)$
$\underset{\text{1st order linear DE}}{\underbrace{\frac{\mathrm{d}y}{\mathrm{d}x}+P\left(x\right)y=Q\left(x\right)}} \left(8\right)$
$\text{Integrating factor}=\exp (\int P(x\left)\mathrm{dx}\right) \left(9\right)$
$\exp (\int -\frac{2}{x}\mathrm{dx}) \left(10\right)$
$=\exp (-2\int \frac{1}{x}\mathrm{dx}) \left(11\right)$
$=\exp (-2\ln \left|x\right|+\text{const}) \left(12\right)$
$=\exp (-2\ln \left|x\right|)\exp \left(\text{const}\right) \left(13\right)$
$=\exp (ln\left|x\right|{)}^{-2}\exp \left(\text{const}\right) \left(14\right)$
$\exp \left(\ln \left|x\right|\right)=\exp \left(\ln \left|x\right|\right) \left(15\right)$
I got the following general solution as I forcefully assumed
$~ x=\exp \left(\ln \left|x\right|\right) ~$
$y=\frac{1}{5}{x}^7+{\text{const}}_1{x}^2 \left(16\right)$

How to solve $\mathbf{\text{x}}\text{'}\text{'}\left(t\right)=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\mathbf{\text{x}}\left(t\right)$
What I'm thinking is to consider the first order version
$\mathbf{\text{x}}\text{'}\left(t\right)=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\mathbf{\text{x}}\left(t\right)$
which I know how to solve, the solution is
$\mathbf{\text{x}}\left(t\right)=\mathbf{\text{c}}\left[\begin{array}{c}{e}^{-t}\\ e^{-t}\end{array}\right]$
How do I use this to solve the second-order equation?

Write out he first four terms of the Maclaurin series of f(x) if 
f(0) = 11, f'(0)=-5, f"(0)=8, f'''(0) = -15