COULD YOU ANSWER THESE PROBLEMS?

1. (6 points) Consider the matrix A = 4 The only eigenvalue for the matrix is A = -1 and the only linearly independent eigenvector is v1 = [1,2]". Find the solution to the initial value problem x' = Ax, x(0) = [2,3]"2. A general solution of x' = Ax is given by x = Cyet _ + Caett (a) (5 points) Sketch the half-line solutions generated by each exponential term of the solution. Then, sketch a rough approximation of a solution in each region determined by the half-line solutions. Use arrows to indicate the direction of motion on all solutions. (b) (1 point) The equilibrium at the origin is best classified as a (circle one): . nodal sink, . saddle point, . spiral source, . nodal source, . spiral sink, . none of these. 3. (2 points) Let A = 6 2 -5 2. Classify the equilibrium point of the system x' = Ax based on the position of (T, D) in the trace-determinant plane. Do NOT solve the system!4. For a real valued parameter b, consider the initial value problem x = 4 x x(0) = x0- This system has an equilibrium at the origin. For each of the following types of equilibria, determine the values of b so that the equilibrium at the origin is of that type. If there are no such b, state so. (a) (1 points) T =. D = (b) (1 point) The equilibrium is a spiral sink. (c) (1 point) The equilibrium is a spiral source. (d) (1 point) The equilibrium is a nodal sink. (e) (1 point) The equilibrium is a saddle. (f) (1 point) For what values of b are all solutions guaranteed to approach the equilibrium at the origin as t -+ Do