solving system of differential equations and eigenvalues

(See attached file for full problem description)
1) The slope field for the system
dx/dt = 2x + 6y
dy/dt = 2x - 2y
is shown to the right
a) determine the type of the equilibrium point at the origin.
b) calculate all straight-line solution.

2) show that a matrix of the form A =(a b; -b a) with b!=0 must have complex eigenvalues.

3) Let A = (a b; c d), define the trace of A to be tr(A) = a +d. show that A has only one eignevlue if and only if
(tr(A))^2 - 4det(A) = 0

The population sizes of a prey, X, and a predator, Y (measured in thousands) are given by x and y, respectively. They are governed by the diﬀerential equations
ẋ = −pxy + qx and ẏ = rxy - sy (where p, q, r and s are positive constants (p ≠ r).
In the absence of species Y (i.e. y = 0), how would I ﬁnd a solution

How do I express the following inhomogeneous system of first-order differentialequations for x(t) and y(t) in matrix form?
(see the attachment for the full question)
x = -2x - y + 12t + 12,
y = 2x - 5y - 5
How do I express the corresponding homogeneous system of differentialequations, also in matrix form?
How do I fin

1. Complex Exponentials: Simply the following expression and give your answer both in polar and rectangular form.
o c=3ejπ/4+4e−jπ/2
2. Difference Equations: Solve the following difference equation using recursion by hand (for n=0 to n=4)
o y[n] + 0.5y[n-1] = 2x[n-1]; x[n] = δ[n], y[-1] = 0
3. DifferentialEquations

See the attached file, please.
Given the following matrices A, B, and C compute the eigenvalue and eigenvector for each matrix.
A =(-2, 6, 6, 3)
B= (4, 2, -1, 6)
C= (0, -1, 1, 2)

The volume of two tanks are V1 =100 gallons and V2 = 200 gallons .The inflow and outflow rates of the system are r = 10 gallons per minute. Suppose that the two tanks both contain fresh water initially, but the inflow to tank 1 is brine at 2 pounds per minute, so that 2 pounds of salt flow into tank 1 each minute. Write a matr