1 x 10 5 7 2 x 2 x 3 3 6 3 x 3 x 4 1 4 8 x 4 x 6 2 8

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1. x ′ = 10 5 7 2 x . 2. x ′ = 3 3 6 3 x . 3. x ′ = 4 1 4 8 x . 4. x ′ = 6 2 8 6 x . 5. x ′ = 1 1 1 1 x . 6. x ′ = 4 0 0 4 x . 7. x ′ = 3 2 4 1 x . 8. x ′ = 6 0 5 6 x . 9. (i) For what value(s) of b will the system below have an improper node at (0,0)? (ii) For what value(s) of b will the system below have a spiral point at (0,0)? x ′ = 1 2 5 b x . 10 – 14 Find the critical point of each nonhomogeneous linear system given. Then determine the type and stability of the critical point. 10. x ′ = 3 6 2 1 x + 24 6 . 11. x ′ = 3 0 0 3 x + 3 12 . 12. x ′ = 0 7 6 0 x + 7 3 . 13. x ′ = 1 2 2 3 x + 2 4 . 14. x ′ = 1 3 6 5 x + 0 13 .
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© 2008 Zachary S Tseng D -2 - 21 Answers D-2.1 : 1. Node, asymptotically stable 2. Center, stable 3. Improper node, unstable 4. Node, unstable 5. Spiral point, asymptotically stable 6. Proper node (or start point), asymptotically stable 7. Saddle point, unstable 8. Improper node, unstable 9. (i) b = 9/2, (ii) b < −9/2 10. (−2, 4) is a saddle point, unstable. 11. (−4, 1) is a proper node (star point), unstable. 12. (1, −0.5) is a center, stable. 13. (0, 2) is an improper node, asymptotically stable 14. (−1, −3) is a spiral point, unstable.
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© 2008 Zachary S Tseng D -2 - 22 Nonlinear Systems Consider a nonlinear system of differential equations: x ′ = F ( x , y ) y ′ = G ( x , y ) Where F and G are functions of two variables: x = x ( t ) and y = y ( t ); and such that F and G are not both linear functions of x and y . Unlike a linear system, a nonlinear system could have none, one, two, three, or any number of critical points. Like a linear system, however, the critical points are found by setting x ′ = y ′ = 0, and solve the resulting system 0 = F ( x , y ) 0 = G ( x , y ) Any and every solution of this system of algebraic equations is a critical point of the given system of differential equations. Since there might be multiple critical points present on the phase portrait, each trajectory could be influenced by more than one critical point. This results in a much more chaotic appearance of the phase portrait. Consequently, the type and stability of each critical point need to be determined locally (in a small neighborhood on the phase plane around the critical point in question) on a case-by-case basis. Without detailed calculation, we could estimate (meaning, the result is not necessarily 100% accurate) the type and stability by a little bit of multi-variable calculus. We will approximate the behavior of the nearby trajectories using the linearizations (i.e. the tangent approximations) of F and G about each critical point. This converts the nonlinear system into a linear system whose phase
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