solve completely the system of constant coefficient, linear, nonhomogeneous differential equations, using eigenvalues and eigenvectors, or particular and general solutions. 3 1 -2 z' (t) with r(0) = 5, y(0) = 0, and 2(0) = 1. 2 4 -4 2 x(t) 3) (66 1 -1 + 2e-t 3t 2t

x = t°x1 + 2x2 + sec(t), x, = sin(t) x1 + tx2 – 3. %3D This system of linear differential equations can be put in the form x = P(t)x + g(t). Determine P(t) and g(t). help (formulas) help (matrices) P(t) : help (formulas) help (matrices) g(t):

SOLVE STEP BY STEP BY HAND (LEGIBLE) Determine the STABILITY of the following systems of differential equations: = 4 -1 X. **()* e) X f) X 1 2 0 3 0x - (21) x 100 X.

Consider a system of differential equations describing the progress of a disease in a population, given by In our particular case, this is: a) Find the nullclines (simplest form) of this system of differential equations. The x-nullcline is y = x = 3-3xy - 1x y = 3xy - 2y where x (t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in units of 1,000 individuals. The y-nullclines are y = and x = b) There are two equilibrium points that are biologically meaningful (i.e. whose coordinates are non-negative). They are (x1,9₁) and (x2, y2), where we order them so that x1 < x2. The equilibrium with the smaller x-coordinate is (x₁, y₁) : The equilibrium with the larger x-coordinate is (x2, y₂) = where A = 0 sin (a) c) The linearization of the system of differential equations at the equilibrium (ï1, Y₁) gives a system of the form (+) ¹ (~), f əx ∞ a Ω = A x' G E = F(x, y) for a vector-valued function…