05 1 15 2 25 3 35 4 45 5 8 6 4 2 2 4 6 8 x 10 4 t

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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -8 -6 -4 -2 0 2 4 6 8 x 10 4 t x(t),y(t) x(t) y(t) Fig. 2.8. Plots of solutions of Example 2.6 for several initial conditions. Example 2.7. Center x = y y = x. (2.21) This system is a simple, coupled system. Neither equation can be solved without some information about the other unknown function. However, we can differentiate the first equation and use the second equation to obtain x ′′ + x = 0 . We recognize this equation from the last chapter as one that appears in the study of simple harmonic motion. The solutions are pure sinusoidal oscilla- tions: x ( t ) = c 1 cos t + c 2 sin t, y ( t ) = c 1 sin t + c 2 cos t. In the phase plane the trajectories can be determined either by looking at the direction field, or solving the first order equation
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34 2 Systems of Differential Equations -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x(t) y(t) Fig. 2.9. Phase portrait for Example 2.6, an unstable node or source. dy dx = x y . Performing a separation of variables and integrating, we find that x 2 + y 2 = C. Thus, we have a family of circles for C> 0 . (Can you prove this using the gen- eral solution?) Looking at the results graphically in Figures 2.10-2.11 confirms this result. This type of point is called a center . Example 2.8. Focus (spiral) x = αx + y y = x. (2.22) In this example, we will see an additional set of behaviors of equilibrium points in planar systems. We have added one term, αx, to the system in Ex- ample 2.7. We will consider the effects for two specific values of the parameter: α = 0 . 1 , 0 . 2 . The resulting behaviors are shown in the remaining graphs. We see orbits that look like spirals. These orbits are stable and unstable spirals (or foci , the plural of focus.) We can understand these behaviors by once again relating the system of first order differential equations to a second order differential equation. Using our usual method for obtaining a second order equation form a system, we find that x ( t ) satisfies the differential equation
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2.2 Equilibrium Solutions and Nearby Behaviors 35 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 -2 -1 0 1 2 3 t x(t),y(t) x(t) y(t) Fig. 2.10. Plots of solutions of Example 2.7 for several initial conditions. -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x(t) y(t) Fig. 2.11. Phase portrait for Example 2.7, a center. x ′′ αx + x = 0 . We recall from our first course that this is a form of damped simple harmonic motion . We will explore the different types of solutions that will result for various α ’s.
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36 2 Systems of Differential Equations 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -4 -3 -2 -1 0 1 2 3 4 t x(t),y(t) x(t) y(t) Fig. 2.12. Plots of solutions of Example 2.8 for several initial conditions with α = 0 . 1. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 -2 -1 0 1 2 3 t x(t),y(t) x(t) y(t) Fig. 2.13. Plots of solutions of Example 2.8 for several initial conditions with α = 0 . 2. The characteristic equation is r 2 αr +1 = 0 . The solution of this quadratic equation is r = α ± α 2 4 2 .
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2.2 Equilibrium Solutions and Nearby Behaviors 37 There are five special cases to consider as shown below.
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