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The general solution of this first order differential

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The general solution of this first order differential equation is found using separation of variables as x = Ay 2 for A an arbitrary constant. Plots of these solutions in the phase plane are given in Figure 2.2. [Note that this is the same form for the orbits that we had obtained above by eliminating t from the solution of the system.] -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x(t) y(t) Fig. 2.2. Orbits for Example 2.3. Once one has solutions to differential equations, we often are interested in the long time behavior of the solutions. Given a particular initial condition ( x 0 ,y 0 ), how does the solution behave as time increases? For orbits near an equilibrium solution, do the solutions tend towards, or away from, the equi- librium point? The answer is obvious when one has the exact solutions x ( t ) and y ( t ). However, this is not always the case.
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2.2 Equilibrium Solutions and Nearby Behaviors 29 Let’s consider the above example for initial conditions in the first quadrant of the phase plane. For a point in the first quadrant we have that dx/dt = 2 x< 0 , meaning that as t → ∞ ,x ( t ) get more negative. Similarly, dy/dt = y< 0 , indicates that y ( t ) is also getting smaller for this problem. Thus, these orbits tend towards the origin as t → ∞ . This qualitative information was obtained without relying on the known solutions to the problem. Direction Fields: Another way to determine the behavior of our system is to draw the direction field. Recall that a direction field is a vector field in which one plots arrows in the direction of tangents to the orbits. This is done because the slopes of the tangent lines are given by dy/dx. For our system (2.5), the slope is dy dx = ax + by cx + dy . In general, for nonautonomous systems, we obtain a first order differential equation of the form dy dx = F ( x,y ) . This particular equation can be solved by the reader. See homework problem 2.2. Example 2.4. Draw the direction field for Example 2.3. We can use software to draw direction fields. However, one can sketch these fields by hand. we have that the slope of the tangent at this point is given by dy dx = y 2 x = y 2 x . For each point in the plane one draws a piece of tangent line with this slope. In Figure 2.3 we show a few of these. For ( x,y ) = (1 , 1) the slope is dy/dx = 1 / 2 . So, we draw an arrow with slope 1 / 2 at this point. From system (2.17), we have that x and y are both negative at this point. Therefore, the vector points down and to the left. We can do this for several points, as shown in Figure 2.3. Sometimes one can quickly sketch vectors with the same slope. For this example, when y = 0, the slope is zero and when x = 0 the slope is infinite. So, several vectors can be provided. Such vectors are tangent to curves known as isoclines in which dy dx =constant. It is often difficult to provide an accurate sketch of a direction field. Com- puter software can be used to provide a better rendition. For Example 2.3 the direction field is shown in Figure 2.4. Looking at this direction field, one can begin to “see” the orbits by following the tangent vectors.
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30 2 Systems of Differential Equations Fig. 2.3.
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