Consider the linear system dx/dt = a11x + a12y, dy/dt = a21x + a22y, where a11, ...
Chapter 9, Problem 20(choose chapter or problem)
Consider the linear system dx/dt = a11x + a12y, dy/dt = a21x + a22y, where a11, ... , 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 the coefficient 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 obtained by studying the eigenvalues r1 and r2. It may also be helpful to establish, and then to use, the relations r1r2 = q and r1 + r2 = p.
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