Suppose the model in is modified so that air resistance is proportional to v2 , that is

In Problems 14 reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve.

In Problems 14 reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve.

In Problems 14 reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve.

In Problems 14 reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve.

In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

In Problems 13 and 14 the given figure represents the graph of f(y) and f(x), respectively. By hand, sketch a direction field over an appropriate grid for dydx f(y) (Problem 13) and then for dydx f(x) (Problem 14). y(

In Problems 13 and 14 the given figure represents the graph of f(y) and f(x), respectively. By hand, sketch a direction field over an appropriate grid for dydx f(y) (Problem 13) and then for dydx f(x) (Problem 14). y(

In parts (a) and (b) sketch isoclines f(x, y) c (see the Remarks on page 38) for the given differential equation using the indicated values of c. Construct a direction fiel over a grid by carefully drawing lineal elements with the appropriate slope at chosen points on each isocline. In each case, use this rough direction field to sketch an approximate solution curve for the IVP consisting of the DE and the initial condition y(0) 1. (a) dydx x  y; c an integer satisfying 5

(a) Consider the direction fiel of the differential equation dydx x( y  4)2  2, but do not use technology to obtain it. Describe the slopes of the lineal elements on the lines x 0, y 3, y 4, and y 5. (b) Consider the IVP dydx x(y  4)2  2, y(0) y0, where y0 4. Can a solution y(x) :  as x : ? Based on the information in part (a), discuss. c 1 4,

For a first-orde DE dydx f(x, y) a curve in the plane define by f(x, y) 0 is called a nullcline of the equation, since a lineal element at a point on the curve has zero slope. Use computer software to obtain a direction fiel over a rectangular grid of points for dydx x2  2y, and then superimpose the graph of the nullcline over the direction field. Discuss the behavior of solution curves in regions of the plane defined by and by . Sketch some approximate solution curves. Try to generalize your observations. 18.

(a) Identify the nullclines (see Problem 17) in Problems 1, 3, and 4. With a colored pencil, circle any lineal elements in Figures 2.1.12, 2.1.14, and 2.1.15 that you think may be a lineal element at a point on a nullcline. (b) What are the nullclines of an autonomous first-orde DE?

Consider the autonomous first-order differential equation dydx y  y3 and the initial condition y(0) y0. By hand, sketch the graph of a typical solution y(x) when y0 has the given values. (a) y0 1 (b) 0 y0 1 (c) 1 y0 0 (d) y0 1 20

Consider the autonomous first-orde differential equation dydx y2  y4 and the initial condition y(0) y0. By hand, sketch the graph of a typical solution y(x) when y0 has the given values. (a) y0 1 (b) 0 y0 1 (c) 1 y0 0 (d) y0 1 In

In Problems 2128 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

In Problems 2128 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

In Problems 2128 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

In Problems 2128 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

In Problems 2128 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

In Problems 2128 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

In Problems 2128 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

In Problems 2128 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

In Problems 29 and 30 consider the autonomous differential equation dydx f(y), where the graph of f is given. Use the graph to locate the critical points of each differential d

In Problems 29 and 30 consider the autonomous differential equation dydx f(y), where the graph of f is given. Use the graph to locate the critical points of each differential d

Consider the autonomous DE dydx (2)y  sin y. Determine the critical points of the equation. Discuss a way of obtaining a phase portrait of the equation. Classify the critical points as asymptotically stable, unstable, or semi-stable. 3

A critical point c of an autonomous first-order DE is said to be isolated if there exists some open interval that contains c but no other critical point. Can there exist an autonomous DE of the form given in (2) for which every critical point is nonisolated? Discuss; do not think profound thoughts.

Suppose that y(x) is a nonconstant solution of the autonomous equation dydx f ( y) and that c is a critical point of the DE. Discuss: Why cant the graph of y(x) cross the graph of the equilibrium solution y c? Why cant f ( y) change signs in one of the subregions discussed on page 39? Why cant y(x) be oscillatory or have a relative extremum (maximum or minimum)? 3

Suppose that y(x) is a solution of the autonomous equation dydx f(y) and is bounded above and below by two consecutive critical points c1 c2, as in subregion R2 of Figure 2.1.6(b). If f(y) 0 in the region, then limx: y(x) c2. Discuss why there cannot exist a number L c2 such that limx: y(x) L. As part of your discussion, consider what happens to y (x) as x : . 2.1

Using the autonomous equation (2), discuss how it is possible to obtain information about the location of points of inflection of a solution curve

Consider the autonomous DE dydx y2  y  6. Use your ideas from Problem 35 to find intervals on the y-axis for which solution curves are concave up and intervals for which solution curves are concave down. Discuss why each solution curve of an initial-value problem of the form dydx y2  y  6, y(0) y0, where 2 y0 3, has a point of inflection with the same y-coordinate. What is that y-coordinate? Carefully sketch the solution curve for which y(0) 1. Repeat for y(2) 2. 37. Su

Suppose the autonomous DE in (2) has no critical points. Discuss the behavior of the solutions.

The differential equation in Example 3 is a well-known population model. Suppose the DE is changed to , where a and b are positive constants. Discuss what happens to the population P as time t increases.

Another population model is given by , where h and k are positive constants. For what initial values P(0) P0 does this model predict that the population will go extinct?

In Section 1.3 we saw that the autonomous differential equation , where k is a positive constant and g is the acceleration due to gravity, is a model for the velocity v of a body of mass m that is falling under the influenc of gravity. Because the term kv represents air resistance, the velocity of a body falling from a great height does not increase without bound as time t increases. Use a phase portrait of the differential equation to fin the limiting, or terminal, velocity of the body. Explain your reasoning.

Suppose the model in Problem 40 is modified so that air resistance is proportional to v2 , that is, . See Problem 17 in Exercises 1.3. Use a phase portrait to find the terminal velocity of the body. Explain your reasoning.

When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled by the autonomous differential equation where k 0 is a constant of proportionality and  0. Here X(t) denotes the number of grams of the new compound formed in time t. (a) Use a phase portrait of the differential equation to predict the behavior of X(t) as t : . dX dt k(  X)(  X), (b) Consider the case when  . Use a phase portrait of the differential equation to predict the behavior of X(t) as t :  when X(0) . When X(0) . (c) Verify that an explicit solution of the DE in the case when k 1 and  is X(t)   1(t  c). Find a solution that satisfies X(0) 2. Then fin a solution that satisfies X(0) 2. Graph these two solutions. Does the behavior of the solutions as t :  agree with your answers to part (b)? Solution b