Direction Fields

8 Key excerpts on "Direction Fields"

• 

  eBook - ePub

  Elementary Differential Equations

  Applications, Models, and Computing

  • Charles Roberts(Author)
  • 2018(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 1 we gave examples of initial value problems with no solution, with a unique solution, and with multiple solutions. Later, we will state a theorem which will guarantee the existence of a solution to an initial value problem of the form (3) and we will state a second theorem which will guarantee the existence and uniqueness of a solution.

   

  2.1. Direction Fields

  First, let us examine the geometric significance of the differential equation (1)

  y ′

  = f

  ( x , y )

  . At each point (x , y ) in the xy -plane for which the function f is defined, the differential equation defines a real value, f (x , y ). This value is the slope of the tangent line to every solution of the differential equation which passes through the point (x , y ). Thus, the differential equation specifies the direction that a solution must have at every point (x , y ) in the domain of f . Imagine passing a short line segment of slope f (x , y ) through each point (x , y ) in the domain of f . The set of all such line segments is called the direction field of the differential equation

  y ′

  = f

  ( x , y )

  . Usually, the domain of f contains an infinite number of points; and, therefore, we cannot possibly draw the direction field. Instead, we choose some rectangle

  R = { ( x , y ) | Xmin ≤ x ≤ Xmax and Ymin ≤ y ≤ Ymax }

  which contains points of the domain of f ; we select a set of points

  (

  x i

  ,

  y i

  )

  contained in R ; and for those points

  (

  x i

  ,

  y i

  )

  in the domain of f , we construct a short line segment at

  (

  x i

  ,

  y i

  )

  with slope

  f (

  x i

  ,

  y i

  )

  . We will call a graph constructed in this manner the direction field of

  y ′

  = f

  ( x , y )

  in the rectangle R . The direction field indicates subregions in R in which solutions are increasing and decreasing, it often reveals maxima and minima of solutions in R , it sometimes indicates the asymptotic behavior of solutions, and it illustrates the dependence of solutions on the initial conditions. Let (x , y ) be a fixed point in the rectangle R at which f (x , y

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  eBook - ePub

  Applied Differential Equations with Boundary Value Problems

  • Vladimir Dobrushkin(Author)
  • 2017(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  1.5 Direction Fields

  A geometrical viewpoint is particularly helpful for the first order equation

  y ′

  = f

  ( x , y )

  . The solutions of this equation form a family of curves in the xy-plane. At any point

  ( x , y )

  , the slope dy /dx of the solution y (x ) at that point is given by

  f ( x , y )

  . We can indicate this by drawing a short line segment (or arrow) through the point

  ( x , y )

  with the slope

  f ( x , y )

  . The collection of all such line segments at each point

  ( x , y )

  of a rectangular grid of points is called a

  direction field or a slope field of the differential equation

  y ′

  = f

  ( x , y )

  .

  By increasing the density of arrows, it would be possible, in theory at least, to approach a limiting curve, the coordinates and slope of which would satisfy the differential equation at every point. This limiting curve—or rather the relation between x and y that defines a function y (x ) —is a solution of

  y ′

  = f

  ( x , y )

  . Therefore the direction field gives the “flow of solutions.” Integral curves obtained from the general solution are all different: there is precisely one solution curve that passes through each point

  ( x , y )

  in the domain of

  f ( x , y )

  . They might be touched by the singular solutions (if any) forming the envelope of a family of integral curves. At each of its points, the envelope is tangent to one of integral curves because they share the same slope.

  Direction Fields can be plotted for differential equations even if they are not necessarily written in the normal form. If the derivative y ′ is determined uniquely from the general equation

  F ( x , y ,

  y ′

  ) = 0

  , the direction field can be obtained for such an equation. However, if the equation

  F ( x , y ,

  y ′

  ) = 0

  defines multiple values for y ′ , then at every such point we would have at least two integral curves with distinct slopes.

  Example 1.5.1: Let us consider the differential equation not in the normal form:

  (1.5.1)

  x

  y ′

  2

  - 2 y

  y ′

  + x = 0 .

  At every point

  ( x , y )

  such that y 2  ≥ x 2 we can assign to y ′

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  eBook - PDF

  Boyce's Elementary Differential Equations and Boundary Value Problems

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  A direction field drawn on a fairly fine grid gives a good picture of the overall behavior of solutions of a differential equation. Usually a grid consisting of a few hundred points is sufficient. The construction of a direction field is often a useful first step in the investigation of a differential equation. Two observations are worth particular mention. First, in constructing a direction field, we do not have to solve equation (6); we just have to evaluate the given function f (t, y) many times. Thus Direction Fields can be readily constructed even for equations that may be quite difficult to solve. Second, repeated evaluation of a given function and drawing a direction field are tasks for which a computer or other computational or graphical aid are well suited. All the Direction Fields shown in this book, such as the one in Figures 1.1.2 and 1.1.3, were computer generated. Field Mice and Owls. Now let us look at another, quite different example. Consider a population of field mice that inhabit a certain rural area. In the absence of predators we assume that the mouse population increases at a rate proportional to the current population. This assumption is not a well-established physical law (as Newton’s law of motion is in Example 1), but it is a common initial hypothesis 1 in a study of population growth. If we denote time by t and the mouse population at time t by p(t), then the assumption about population growth can be expressed by the equation dp dt = rp, (7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A better model of population growth is discussed in Section 2.5. 1.1 Some Basic Mathematical Models; Direction Fields 5 where the proportionality factor r is called the rate constant or growth rate. To be specific, suppose that time is measured in months and that the rate constant r has the value 0.5/month.

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  eBook - PDF

  A Course in Differential Equations with Boundary Value Problems

  • Stephen A. Wirkus, Randall J. Swift, Ryan Szypowski(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  2.1. Direction Fields—the Geometry of ODEs 71 Notice how the solution passes tangentially through the direction field. That is, each line segment is tangent to the solution in the direction field. Now, of course, we could have easily calculated the solution to dy dx = y, y(0) = 1 as y(x) = e x and we should observe that this is the curve plotted. Let’s try this method again for an equation for which we don’t know the solution. Example 2 Draw the direction field for the equation dy dx = x 2 + y 2 and sketch several solution curves that pass through the direction field. Solution The direction field for dy dx = x 2 + y 2 is shown in Figure 2.2a. The reader should check a few pairs of points on the graph to make sure the direction field is correct, e.g., check the pairs (3, 0) and (0, 1 2 ). One could also observe that the isoclines are given by x 2 + y 2 = k, which are circles of radius √ k, which indicate that the line segments have the same slope along any circle in the x-y plane. On this direction field, we can plot several graphs of y(x), shown in Figure 2.2b. (a) (b) FIGURE 2.2: The direction field for dy dx = x 2 + y 2 is shown in (a). Graphs of several solutions y(x) on the direction field for dy dx = x 2 + y 2 are shown in (b). We note two points that should be obvious: 1. The finer the mesh of the grid for the representation of the slope field, the better the approximate solution curve we are able to draw. This is the same as saying the tangent line is a good approximation to a curve close to the point of tangency. And the more points we have, the better we can sketch this approximation. 72 Chapter 2. First-Order Geometrical and Numerical Methods 2. Drawing a direction field by hand is tedious—it is best to use a computer. It is, however, essential that we check a few points of the direction field (generated by the computer) by hand. As much as we would like computers to always give us the answers we want, this will never be the case.

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  eBook - PDF

  Differential Equations

  An Introduction to Modern Methods and Applications

  • James R. Brannan, William E. Boyce(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  (4a) vary with time. On a printed copy of a direction field we can even sketch (approximately) graphs of solutions by drawing curves that are always tangent to line segments in the direction field. Thus the general geometric behavior of the integral curves can be inferred from the direction field in Figure 1.2.6. This approach can be applied equally well to the more general Eq. (4), where the param- eters k and T 0 are unspecified positive numbers. The conclusions are essentially the same. The equilibrium solution of Eq. (4) is u = T 0 . Solutions below the equilibrium solution in- crease with time, those above it decrease with time, and all other solutions approach the equilibrium solution as t becomes large. The connection between integral curves and Direction Fields is an important concept for understanding how the right side of a differential equation, such as u ′ = −k(u − T 0 ), 1.2 Qualitative Methods: Phase Lines and Direction Fields 23 0.5 0 1 1.5 2 2.5 t 50 45 55 60 65 70 u FIGURE 1.2.6 Direction field and equilibrium solution u = 60 for u ′ = −1.5(u − 60). determines the behavior of solutions and gives rise to the integral curves. However, using modern software packages, it is just as easy to plot the graphs of numerical approximations to solutions as it is to draw Direction Fields. We will frequently do this because the behavior of solutions of a first order equation is usually made most clear by overlaying the direction field with a representative set of integral curves, as shown in Figure 1.2.7. Such a sampling of integral curves facilitates visualization of the many other integral curves determined by the direction field generated by the right side of the differential equation. 0.5 0 1 1.5 2 2.5 50 45 55 60 65 70 u t FIGURE 1.2.7 Direction field for u ′ = −1.5(u − 60) overlaid with the integral curves shown in Figure 1.1.2.

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  eBook - PDF

  Single Variable Calculus

  Concepts and Contexts, Enhanced Edition

  • James Stewart(Author)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  SOLUTION (a) If we put , , and in Equation 1, we get The direction field for this differential equation is shown in Figure 10. (b) It appears from the direction field that all solutions approach the value 5 A, that is, (c) It appears that the constant function is an equilibrium solution. Indeed, we can verify this directly from the differential equation . If , then the left side is and the right side is . (d) We use the direction field to sketch the solution curve that passes through , as shown in red in Figure 11. Notice from Figure 10 that the line segments along any horizontal line are parallel. That is because the independent variable t does not occur on the right side of the equation FIGURE 11 0 t 1 I 2 3 2 4 6 0, 0 15 3 5 0 dI dt 0 I t 5 dI dt 15 3I I t 5 lim t l I t 5 FIGURE 10 0 t 1 I 2 3 2 4 6 dI dt 15 3I or 4 dI dt 12I 60 E t 60 R 12 L 4 I 0 0 t 0 12 EXAMPLE 2 v SECTION 7.2 Direction Fields AND EULER’S METHOD 503 . In general, a differential equation of the form in which the independent variable is missing from the right side, is called autonomous. For such an equation, the slopes corresponding to two different points with the same -coordinate must be equal. This means that if we know one solution to an autonomous differential equation, then we can obtain infinitely many others just by shifting the graph of the known solution to the right or left. In Figure 11 we have shown the solutions that result from shifting the solution curve of Example 2 one and two time units (namely, seconds) to the right. They correspond to closing the switch when or . Euler’s Method The basic idea behind Direction Fields can be used to find numerical approximations to solutions of differential equations. We illustrate the method on the initial-value problem that we used to introduce Direction Fields: The differential equation tells us that , so the solution curve has slope 1 at the point . As a first approximation to the solution we could use the linear approx- imation .

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  eBook - PDF

  Economic Dynamics

  Phase Diagrams and their Economic Application

  • Ronald Shone(Author)
  • 2002(Publication Date)
  • Cambridge University Press

    (Publisher)

  Systems of first-order differential equations 189 So long as solutions exist, then the packages will solve the system of equations. Thus, the system of three equations with initial values: x ( t ) = x ( t ) y ( t ) = x ( t ) + 3 y ( t ) − z ( t ) z ( t ) = 2 y ( t ) + 3 x ( t ) x (0) = 1 , y (0) = 1 , z (0) = 2 (4.36) can be solved in a similar manner with no difficulty. In the case of nonlinear systems of differential equations, or where no explicit solution can be found, then it is possible to use the NDSolve command in Math-ematica and the dsolve(. . . , numeric) command in Maple to obtain numerical approximations to the solutions. These can then be plotted. But often more in-formation can be obtained from direction field diagrams and phase portraits. A direction field shows a series of small arrows that are tangent vectors to solutions of the system of differential equations. These highlight possible fixed points and most especially the flow of the system over the plane. A phase portrait, on the other hand, is a sample of trajectories (solution curves) for a given system. Figure 4.36(a) shows a direction field and figure 4.36(b) a phase portrait. In many instances Direction Fields and phase portraits are combined on the one diagram – as we have done in many diagrams in this chapter. The phase portrait can be derived by solving a system of differential equations, if a solution exists. Where no known solution exists, trajectories can be obtained by using numerical Figure 4.36. 190 Economic Dynamics solutions. These are invariably employed for systems of nonlinear differential equation systems. 4.12.1 Direction Fields and phase portraits with Mathematica Direction Fields in Mathematica are obtained using the PlotVectorField command. In order to use this command it is first necessary to load the PlotField package.

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  eBook - ePub

  Introduction to Differential Equations with Dynamical Systems

  • Stephen L. Campbell, Richard Haberman(Authors)
  • 2011(Publication Date)
  • Princeton University Press

    (Publisher)

  x are important? What is the behavior of the differential equation near these important points? Why do the solutions of the differential equations behave the way they are observed numerically to act? The direction field method is basically a numerical method and, while useful, it is far from the best numerical method. Section 1.5 will discuss how to view it as a numerical method. We often need to get a better understanding of the solutions of differential equations than can be obtained this way. That is one reason we wish to study differential equations further.

  Figure 1.4.5 Direction field and solutions of (9), = x 2 − t .

  Exercises _______________________________________________

  In Exercises 1–6, the direction field of the given differential equations has been plotted by a computer. Sketch some solutions of the differential equation. (Hint : First copy or trace the figure.)

  1. = x (1 − x 2 ).

  2. = t (1 − t 2 ).

  3. = x (t − x ).

  4. = t (x − t ).

  5. = t 2 − x 2 .

  6. = t 2 − x ).

  1.4.1 Existence and Uniqueness

  Often the first-order differential equation = f (t, x ) cannot be solved by simple integration. It is then important to know when there are solutions. As with the previous elementary examples, the differential equation usually has many solutions, of which only one will satisfy given initial conditions. A more precise statement of the mathematical result which guarantees this is the following.

  THEOREM 1.4.1 Basic existence and uniqueness theorem for the initial value problem for first-order differential equations. There exists a unique solution to the differential equation

  which satisfies given initial conditions

  if both the function f (t, x ) and its partial derivative are continuous functions of t and x at and near the initial point t = t 0 , x = x 0 .

  In most cases of practical interest, the continuity conditions are satisfied for most values of (t 0 , x 0 ), so that there exists a unique solution to the initial value problem for most initial conditions.

  If the conditions of the basic existence and uniqueness theorem are not met at a point (t 0 , x 0 ), then solutions may not exist at (t 0 , x 0 ), or there may be more than one solution passing through the same point (t 0 , x 0 ), or the solution may not be differentiable at t 0

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