The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases.

Determine whether the given set S is a subspace of the vector space V.
A. V=$P_5$, and S is the subset of $P_5$ consisting of those polynomials satisfying p(1)>p(0).
B. $V=R_3$, and S is the set of vectors $(x_1,x_2,x_3)$ in V satisfying $x_1-6x_2+x_3=5$.
C. $V=R^n$, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=$C^2(I)$, and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.
F. V=$P_n$, and S is the subset of $P_n$ consisting of those polynomials satisfying p(0)=0.
G. $V=M_n(R)$, and S is the subset of all symmetric matrices

I am studying about the linear odes with non-constant coefficients.
I know the first order linear ode with non-constant coefficient
$\begin{array}{}\text{(1)}& y^{{}^{\prime }}(x)+f(x)y(x)=0\end{array}$
has a general solution of the form
$\begin{array}{}\text{(2)}& y=C{e}^{-\int f(x)dx}\end{array}$
However, I am more interested in the case of linear second order odes with non-constant coefficients
$\begin{array}{}\text{(3)}& y^{{}^{″}}(x)+g(x)y^{{}^{\prime }}(x)+f(x)y(x)=0\end{array}$
I know that this equation does not have a closed form solution like (2). However, I am interested in special cases of that.
Questions
1. Consider (3), when $g(x)=0$, then we have
$\begin{array}{}\text{(4)}& y^{{}^{″}}(x)+f(x)y(x)=0\end{array}$
Is Eq.(4) a famous well-known equation? If YES, what is its name?
2. Does (4) have a closed form solution like (2)?
3. Can you name or give me a list of well-known linear second order odes with non-constant coefficients which are not polynomial?
For example, I know Cauchy-Euler, Airy, Bessel, Chebyshev, Laguerre and Legendre equations whose coefficients are polynomials. But I don't know any well-known equation with non-polynomial coefficients.

-4x - 15y = -17
-x + 5y = -13
x = ___, y = ___

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $x_1,x_2$... from left to right.

$\left[\begin{array}{ccccc}1& 0& 0& -7& 8\\ 0& 1& 0& 3& 2\\ 0& 0& 1& 1& -5\end{array}\right]$

Write the vector form of the general solution of the given system of linear equations.
$x_1+2x_2-x_3+x_4=0$

Solve the system by clennaton

$\{\begin{array}{l}2x-y=0\\ 3x-2y=-3\end{array}$
The solution is____

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution.

$\left[\begin{array}{cccc}1& -2& 0& -4\\ 0& 0& 1& 3\\ 0& 0& 0& 0\end{array}\right]$