(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 solution shows how to solve the system of differential equations. Furthermore, it characterizes the equilibrium at the origin.