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In this case we see that all solutions tend towards

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In this case we see that all solutions tend towards the equilibrium point, (0 , 0) . This will be called a stable node , or a sink . Before looking at other types of solutions, we will explore the stable node in the above example. There are several methods of looking at the behavior of solutions. We can look at solution plots of the dependent versus the inde- pendent variables, or we can look in the xy -plane at the parametric curves ( x ( t ) ,y ( t )). Solution Plots: One can plot each solution as a function of t given a set of initial conditions. Examples are are shown in Figure 2.1 for several initial conditions. Note that the solutions decay for large t. Special cases result for various initial conditions. Note that for t = 0 ,x (0) = c 1 and y (0) = c 2 . (Of course, one can provide initial conditions at any t = t 0 . It is generally easier to pick t = 0 in our general explanations.) If we pick an initial condition with c 1 =0, then x ( t ) = 0 for all t . One obtains similar results when setting y (0) = 0 .
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2.2 Equilibrium Solutions and Nearby Behaviors 27 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. 2.1. Plots of solutions of Example 2.3 for several initial conditions. Phase Portrait: There are other types of plots which can provide ad- ditional information about our solutions even if we cannot find the exact solutions as we can for these simple examples. In particular, one can consider the solutions x ( t ) and y ( t ) as the coordinates along a parameterized path, or curve, in the plane: r = ( x ( t ) ,y ( t )) Such curves are called trajectories or orbits . The xy -plane is called the phase plane and a collection of such orbits gives a phase portrait for the family of solutions of the given system. One method for determining the equations of the orbits in the phase plane is to eliminate the parameter t between the known solutions to get a relation- ship between x and y. In the above example we can do this, since the solutions are known. In particular, we have x = c 1 e 2 t = c 1 parenleftbigg y c 2 parenrightbigg 2 Ay 2 . Another way to obtain information about the orbits comes from noting that the slopes of the orbits in the xy -plane are given by dy/dx. For au- tonomous systems, we can write this slope just in terms of x and y. This leads to a first order differential equation, which possibly could be solved analyt- ically, solved numerically, or just used to produce a direction field . We will see that direction fields are useful in determining qualitative behaviors of the solutions without actually finding explicit solutions. First we will obtain the orbits for Example 2.3 by solving the corresponding slope equation. First, recall that for trajectories defined parametrically by x = x ( t ) and y = y ( t ), we have from the Chain Rule for y = y ( x ( t )) that
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28 2 Systems of Differential Equations dy dt = dy dx dx dt . Therefore, dy dx = dy dt dx dt . (2.18) For the system in (2.17) we use Equation (2.18) to obtain the equation for the slope at a point on the orbit: dy dx = y 2 x .
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