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Consider the linear system dx dt = a11x + a12 y, dy dt = a21x + a22 y, where a11, a12
Chapter 9, Problem 17(choose chapter or problem)
Consider the linear system dx dt = a11x + a12 y, dy dt = a21x + a22 y, where a11, a12, a21, and a22 are real-valued 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 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. 1
Questions & Answers
QUESTION:
Consider the linear system dx dt = a11x + a12 y, dy dt = a21x + a22 y, where a11, a12, a21, and a22 are real-valued 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 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. 1
ANSWER:Step 1 of 9
Consider the linear system is
Where 
This system can also be written as
Let 
