How many non cyclic groups of order 14

The set $B=\{1+t^2,t+t^2,1+2t+t^2\}$ is a basis for ${\displaystyle {\mathbb{P}}_2}$. Find the coordinate vector of ${\displaystyle p\left(t\right)=1+4t+7t^2}$ relative to B.

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
${\displaystyle \lambda 1=-1\to \left\{\begin{array}{cc}1& 1\end{array}\right\},\lambda 2=2\to \left\{\begin{array}{cc}1& -1\end{array}\right\}}$

Suppose that ${\displaystyle f\left(x\right)=0,125x\text{ for }0<x<4}$. Determine the mean and variance of X.

In studying a reflection-transmission problem involving exotic materials, I have come across the following linear first-order differential equation:
$\begin{array}{}\text{(1)}& A\frac{\mathrm{\partial }}{\mathrm{\partial }t}g(t)+Bg(t)=f(t),\end{array}$
where A and B are constants, g(t) is associated with the reflected wave, and f(t) is a (finite) driving function associated with the incident wave. Both A and B may be positive or negative. I am interested in the behavior of the solution in the limit that $A\to 0$
In studying a reflection-transmission problem involving exotic materials, I have come across the following linear first-order differential equation:A∂∂tg(t)+Bg(t)=f(t),(1)where A and B are constants, g(t) is associated with the reflected wave, and f(t) is a (finite) driving function associated with the incident wave. Both A and B may be positive or negative. I am interested in the behavior of the solution in the limit that A\rightarrow0.
I know there is an exact solution to Eq. (1), which is
$g(t)=C{e}^{-Bt/A}+\frac{1}{A}{\int }_{-\mathrm{\infty }}^t{e}^{-B(t-t^{\prime })/A}f(t^{\prime })d{t}^{\prime },$
where C=0 because g(t)=0 if f(t)=0. However, I do not understand how this exact solution reduces to the case where A=0, which is $g(t)=B^{-1}f(t)$. Any insight would be greatly appreciated.
I've seen a lot of documents discussing asymptotic analyses of linear differential equations, but they all start with second-order equations. Is this because there is inherently problematic with first-order?

If X and Y are random variables and c is any constant, show that $E(cX)=cE(X)$.

Describe the translation of figure ABCD. Use the drop-down menus to explain your answer.
Figure ABCD is translated _____ unit(s) right and ______ unit(s) up.