17.Consider the linear system dxdt=a11x+a12y,dydt=a21x+a22y,dxdt=a11x+a12y,dydt=a21x+a22y, where a11, a12, a21, and a22 are real-valued constants. Let p = a11 + a22, q = a11a22 − a12a21, and Δ = 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 Δ ≥ 0; b.Saddle point if q < 0; c.Spiral point if p ≠ 0 and Δ < 0; d.Center if p = 0 and q > 0. Hint: These conclusions can be reached 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.
17.Consider the linear system dxdt=a11x+a12y,dydt=a21x+a22y,dxdt=a11x+a12y,dydt=a21x+a22y, where a11, a12, a21, and a22 are real-valued constants. Let p = a11 + a22, q = a11a22 − a12a21, and Δ = 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 Δ ≥ 0; b.Saddle point if q < 0; c.Spiral point if p ≠ 0 and Δ < 0; d.Center if p = 0 and q > 0. Hint: These conclusions can be reached 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.
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter11: Systems Of Equations
Section11.5: Cramer's Rule
Problem 35PS
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17.Consider the linear system
dxdt=a11x+a12y,dydt=a21x+a22y,dxdt=a11x+a12y,dydt=a21x+a22y,
where a11, a12, a21, and a22 are real-valued constants. Let p = a11 + a22, q = a11a22 − a12a21, and Δ = 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 Δ ≥ 0;
b.Saddle point if q < 0;
c.Spiral point if p ≠ 0 and Δ < 0;
d.Center if p = 0 and q > 0.
Hint: These conclusions can be reached 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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