A First Course in Differential Equations with Modeling Applications 10th Edition solutions

A First Course in Differential Equations with Modeling Applications 10
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. \(\frac{d y}{d x}=x^{2}-y^{2}\) (a) y(-2) = 1 (b)

A First Course in Differential Equations with Modeling Applications 10
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. \(\frac{d y}{d x}=e^{-0.01 x y^{2}}\) (a) y(-6) = 0

A First Course in Differential Equations with Modeling Applications 10
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. \(\frac{d y}{d x}=1-x y\) (a) y(0) = 0 (b)

A First Course in Differential Equations with Modeling Applications 10
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. \(\frac{d y}{d x}=(\sin x) \cos y\) (a) y(0) = 1

A First Course in Differential Equations with Modeling Applications 10
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. \(y^{\prime}=x\) (a) y(0) = 0 (b) y(0) = -3

A First Course in Differential Equations with Modeling Applications 10
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. \(y^{\prime}=x+y\) (a) y(-2) = 2 (b) y(1) = -3

A First Course in Differential Equations with Modeling Applications 10
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. \(y \frac{d y}{d x}=-x\) (a) y(1) = 1 (b) y(0) = 4

A First Course in Differential Equations with Modeling Applications 10
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. \(\frac{d y}{d x}=\frac{1}{y}\) (a) y(0) = 1 (b) y(-2) = -1

A First Course in Differential Equations with Modeling Applications 10
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. \(\frac{d y}{d x}=0.2 x^{2}+y\) (a) \(y(0)=\frac{1}{2}\) (b) y(2) = -1

A First Course in Differential Equations with Modeling Applications 10
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. \(\frac{d y}{d x}=x e^{y}\) (a) y(0) = -2 (b) y(1) = 2.5

A First Course in Differential Equations with Modeling Applications 10
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. \(y^{\prime}=y-\cos \frac{\pi}{2} x\) (a) y(2) = 2 (b) y(-1) = 0

A First Course in Differential Equations with Modeling Applications 10
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. \(\frac{d y}{d x}=1-\frac{y}{x}\) (a) \(y\left(-\frac{1}{2}\right)=2\)

A First Course in Differential Equations with Modeling Applications 10
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 dy/dx = f(y) (Problem 13) and then for dy/dx = f(x) (Problem 14).

A First Course in Differential Equations with Modeling Applications 10
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 dy/dx = f(y) (Problem 13) and then for dy/dx = f(x) (Problem 14).

A First Course in Differential Equations with Modeling Applications 10
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, u

A First Course in Differential Equations with Modeling Applications 10
(a) Consider the direction fiel of the differential equation \(d y / d x=x(y-4)^{2}-2\), 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 \(d y / d x=x(y-4)^{2}-2\), \(y(0)=y_{0}\), where \(y_{0}=4

A First Course in Differential Equations with Modeling Applications 10
For a first-orde DE dy/dx = 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 \(d y / d x=x^{2}-2 y\), and then

A First Course in Differential Equations with Modeling Applications 10
(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-order DE?

A First Course in Differential Equations with Modeling Applications 10
Consider the autonomous first-order differential equation \(d y / d x=y-y^{3}\) and the initial condition \(y(0)=y_{0}\). By hand, sketch the graph of a typical solution y(x) when \(y_{0}\) has the given values. (a) \(y_{0}>1\) (b) \(0 < \ y_{0} < 1\) (c) \(-

A First Course in Differential Equations with Modeling Applications 10
Consider the autonomous first-order differential equation \(d y / d x=y^{2}-y^{4}\) and the initial condition \(y(0)=y_{0}\). By hand, sketch the graph of a typical solution y(x) when \(y_{0}\) has the given values. (a) \(y_{0}>1\) (b) \(0 < \ y_{0} < 1\) (c)

A First Course in Differential Equations with Modeling Applications 10
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 t

A First Course in Differential Equations with Modeling Applications 10
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 t

A First Course in Differential Equations with Modeling Applications 10
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 t

A First Course in Differential Equations with Modeling Applications 10
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 t

A First Course in Differential Equations with Modeling Applications 10
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 t

A First Course in Differential Equations with Modeling Applications 10
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 t

A First Course in Differential Equations with Modeling Applications 10
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 t

A First Course in Differential Equations with Modeling Applications 10
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 t

A First Course in Differential Equations with Modeling Applications 10
In Problems 29 and 30 consider the autonomous differential equation dy/dx = f(y), where the graph of f is given. Use the graph to locate the critical points of each differential equation. Sketch a phase portrait of each differential equation. By hand, sketch typical solution curves in the subregions

A First Course in Differential Equations with Modeling Applications 10
In Problems 29 and 30 consider the autonomous differential equation dy/dx = f(y), where the graph of f is given. Use the graph to locate the critical points of each differential equation. Sketch a phase portrait of each differential equation. By hand, sketch typical solution curves in the subregions

A First Course in Differential Equations with Modeling Applications 10
Consider the autonomous DE \(d y / d x=(2 / \pi) y-\sin y\). 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.

A First Course in Differential Equations with Modeling Applications 10
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.

A First Course in Differential Equations with Modeling Applications 10
Suppose that y(x) is a non constant solution of the autonomous equation dy/dx = 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)

A First Course in Differential Equations with Modeling Applications 10
Suppose that y(x) is a solution of the autonomous equation dy/dx = f(y) and is bounded above and below by two consecutive critical points \(c_{1} < \ c_{2}\), as in subregion \(R_{2}\) of Figure 2.1.6(b). If f(y) > 0 in the region, then \(\lim _{x \rightarrow \infty} y(x)=c_{2}\). Discuss why there c

A First Course in Differential Equations with Modeling Applications 10
Using the autonomous equation (2), discuss how it is possible to obtain information about the location of points of inflection of a solution curve

A First Course in Differential Equations with Modeling Applications 10
Consider the autonomous DE \(d y / d x=y^{2}-y-6\). 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 \(d y / d x=y^{

A First Course in Differential Equations with Modeling Applications 10
Suppose the autonomous DE in (2) has no critical points. Discuss the behavior of the solutions.

A First Course in Differential Equations with Modeling Applications 10
Population Model The differential equation in Example 3 is a well-known population model. Suppose the DE is changed to \(\frac{d P}{d t}=P(a P-b)\) where a and b are positive constants. Discuss what happens to the population P as time t increases.

A First Course in Differential Equations with Modeling Applications 10
Population Model Another population model is given by \(\frac{d P}{d t}=k P-h\) where h and k are positive constants. For what initial values \(P(0)=P_{0}\) does this model predict that the population will go extinct? Tex

A First Course in Differential Equations with Modeling Applications 10
Terminal Velocity In Section 1.3 we saw that the autonomous differential equation \(m \frac{d v}{d t}=m g-k v\) 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 influence of gravity. Because th

A First Course in Differential Equations with Modeling Applications 10
Suppose the model in Problem 40 is modified so that air resistance is proportional to \(v^{2}\) , that is, \(m \frac{d v}{d t}=m g-k v^{2}\) See Problem 17 in Exercises 1.3. Use a phase portrait to find the terminal velocity of the body. Explain your reasoning.

A First Course in Differential Equations with Modeling Applications 10
Chemical Reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled by the autonomous differential equation \(\frac{d X}{d t}=k(\alpha-X)(\beta-X)\) where k > 0 is a constant of proportionality and \(\beta>\alpha>0\). Here X(t) denotes the