Consider the linear systemdx/dt = a11x + a12y, dy/dt = a21x + a22y,where a11, a12, a21

Chapter 9, Problem 20

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Consider the linear systemdx/dt = a11x + a12y, dy/dt = a21x + a22y,where a11, a12, a21, and a22 are real constants. Let p = a11 + a22, q = a11a22 a12a21, and = p2 4q. Observe that p and q are the trace and determinant, respectively, of thecoefficient matrix of the given system. Show that the critical point (0, 0) is a(a) Node if q > 0 and 0; (b) Saddle point if q < 0;(c) Spiral point if p = 0 and < 0; (d) Center if p = 0 and q > 0.Hint: These conclusions can be reached by studying the eigenvalues r1 and r2. It may alsobe helpful to establish, and then to use, the relations r1r2 = q and r1 + r2 = p.