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Consider the following phase portrait correspondent O a linear system of first order, ,2.. -3.5 Choose the one statement below that is true: -0.5 2.5 3.5 B c. D E. Both eigenvalues of the coefficient matrix are positive. Both eigenvalues of the coefficient matrix are real values. Both eigenvalues of the coefficient matrix are pure imaginary.

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If you check the box "show eigenvalues", then the phase plane plot shows an overlay of the eigenvalues, where the axes are reused to represent the real and imaginary axes of the complex plane. The eigenvalues appear as two points on this complex plane, and will be along the x-axis (the real axis) if the eigenvalues are real.

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that its eigenvalues are = 0.06 ± p.0684i. Thus, the phase diagram here is a spiral sink! To ﬁnd if these spirals are traversed in the clock-wise or counter-clockwise direction we observe that at the point (h,k)=(1,0) the phase curves go in the direction of 0 B @ dh dt dk dt 1 C A = 0.12 0.03!, i.e. counter-clockwise.

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Consider the homogeneous linear first-order system differential equations x'=ax+by y'=cx+dy. which can be written in matrix form as X'=AX, where A is the coefficients matrix. . The following worksheet is designed to analyse the nature of the critical point (when ) and solutions of the linear system X'

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In this type of phase portrait, the trajectories given by the eigenvectors of the negative eigenvalue initially start at in nite-distance away, then move toward and eventually converge at the critical point. The trajectories that represent the eigenvectors of the positive eigenvalue move in exactly the

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In addition, use pplane (http://math.rice.edu/~dfield/dfpp.html) to plot the phase portraits of the following systems. For each system use the phase portrait to guess the nature of the eigenvalues (distinct, repeated, complex) and their sign (for complex eigenvalues guess the sign of the real part).