a. Find the most general real-valued solution to the linear -2 -25 x. 1 -2 system of differential equations ¹ = x₁ (t) x₂ (t) = C1 b. In the phase plane, this system is best described as a O source / unstable node sink / stable node saddle center point / ellipses spiral source spiral sink none of these

a) Find the initial value problem of the system of linear differential equations: [1 -3 x'(t) = 3 х, х(0) — 1 b) Describe the behavior of the solutions as t -→ 0.

Consider the system of non-linear differential equations:  dx/dt = x cos y,  dy/dt = y cos x.  Which option gives the Jacobian matrix for this system?  A. [-y sin x   cos x]     [cos y    -x sin y] B. [-x sin y   cos y]     [cos x    -y sin x] C. [cos x   -y sin x]      [-x sin y   cos y] D. [x cos y  x cos y]    [y cos x   y cos x] E. [cos y   -x sin y]    [-y sin x   cos x]

6.) Give an example of the following. No need to solve. a.) A first-order, linear, non-homogeneous differential equation with non-constant coefficients b.) A second-order, linear, homogeneous system of differential equations with constant coefficients