Problems 20 through 22 deal with the case ∈ = -1, for which the system in (6) becomes

and imply that the three critical points (0,0), (3,0), and (5,2) of (8) are as shown in Fig. 9.3.17-with a nodal sink at the origin, a saddle point on the positive x-axis, and a spiral source at (5,2). In each problem use a graphing calculator or computer system to construct a phase plane portrait for the linearization at the indicated critical point. Do your local portraits look consistent with Fig. 9.3.17?

Show that the coefficient matrix of the linearization x' = -3x, y' = -5y of the system in (8) at (0, 0) has the negative eigenvalues λ₁ = -3 and λ₂ = -5. Hence (0,0) is a nodal sink for (8).

dx dt - 3x + x - xy, dy dt :-5y+xy, (8)