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Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621
BuyFind

Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621

Solutions

Chapter
Section
Chapter 9.3, Problem 28E
Textbook Problem

27–28

(a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation.

(b) Solve the differential equation.

(c) Use the CAS to draw several members of the family of solutions obtained in part (b). Compare with the curves from part (a).

y = x y

Expert Solution
To determine

To draw:

A direction field for the given differential equation and then sketch some solution curves without solving the differential equation.

Explanation of Solution

1) Concept:

Use Mathematica to draw a direction field and then sketch some solution curves with your hand.

2) Given:

y'=xy

3) Calculation:

To draw direction field, use below command in Mathematica

dfield=VectorPlot[{1,x*y},{x,-3,3},{y,-3,3},AxesTrue,AxesLabel{"x","dydx=x*y"},VectorPointsFine,VectorStyle"Segment"]

then output is

Now, draw some solution curves with hand without solving the differential equation is

Conclusion:

The graph of directional field using Mathematica is

And graph of some solution curves with hand without solving the differential equation is

To solve:

The given differential equation y'=xy

Solution:

y=Kex22 & y=0

1) Concept:

To solve the differential equation of the form dydx=gxhy , rewrite the equation in the differential form  hydy=gxdx so that all  y’s are on one side and all  x’s are on the other side

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Chapter 9 Solutions

Calculus (MindTap Course List)
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Ch. 9.1 - Explain why the functions with the given graphs...Ch. 9.1 - The function with the given graph is a solution of...Ch. 9.1 - Match the differential equations with the solution...Ch. 9.1 - Suppose you have just poured a cup of freshly...Ch. 9.1 - Psychologists interested in learning theory study...Ch. 9.1 - Von Bertalanffys equation states that the rate of...Ch. 9.1 - Differential equations have been used extensively...Ch. 9.2 - A direction field for the differential equation...Ch. 9.2 - A direction field for the differential equation...Ch. 9.2 - 36 Match the differential equation with its...Ch. 9.2 - 36 Match the differential equation with its...Ch. 9.2 - 36 Match the differential equation with its...Ch. 9.2 - 36 Match the differential equation with its...Ch. 9.2 - Use the direction field labeled I above to sketch...Ch. 9.2 - Use the direction field labeled III above to...Ch. 9.2 - 910 Sketch a direction field for the differential...Ch. 9.2 - 910 Sketch a direction field for the differential...Ch. 9.2 - 1114 Sketch the direction field of the...Ch. 9.2 - 1114 Sketch the direction field of the...Ch. 9.2 - 1114 Sketch the direction field of the...Ch. 9.2 - 1114 Sketch the direction field of the...Ch. 9.2 - 1516 Use a computer algebra system to draw a...Ch. 9.2 - 1516 Use a computer algebra system to draw a...Ch. 9.2 - Use a computer algebra system to draw a direction...Ch. 9.2 - Make a rough sketch of a direction field for the...Ch. 9.2 - a Use Eulers method with each of the following...Ch. 9.2 - A direction field for a differential equation is...Ch. 9.2 - Use Eulers method with step size 0.5 to compute...Ch. 9.2 - Use Eulers method with step size 0.2 to estimate...Ch. 9.2 - Use Eulers method with step size 0.1 to estimate...Ch. 9.2 - a Use Eulers method with step size 0.2 to estimate...Ch. 9.2 - a Program a calculator or computer to use Eulers...Ch. 9.2 - a Program your computer algebra system, using...Ch. 9.2 - The figure shows a circuit containing an...Ch. 9.2 - In Exercise 9.1.14 we considered a 95C cup of...Ch. 9.3 - 110 Solve the differential equation. dydx=3x2y2Ch. 9.3 - 110 Solve the differential equation. dydx=xyCh. 9.3 - 110 Solve the differential equation. xyy=x2+1Ch. 9.3 - 110 Solve the differential equation. y+xey=0Ch. 9.3 - 110 Solve the differential equation. (ey1)y=2+cosxCh. 9.3 - 110 Solve the differential equation....Ch. 9.3 - 110 Solve the differential equation. ddt=tsecet2Ch. 9.3 - 110 Solve the differential equation....Ch. 9.3 - 110 Solve the differential equation. dpdt=t2pp+t21Ch. 9.3 - 110 Solve the differential equation. dzdt=et+z=0Ch. 9.3 - 1118 Find the solution of the differential...Ch. 9.3 - 1118 Find the solution of the differential...Ch. 9.3 - 1118 Find the solution of the differential...Ch. 9.3 - 1118 Find the solution of the differential...Ch. 9.3 - 1118 Find the solution of the differential...Ch. 9.3 - 1118 Find the solution of the differential...Ch. 9.3 - 1118 Find the solution of the differential...Ch. 9.3 - 1118 Find the solution of the differential...Ch. 9.3 - Find an equation of the curve that passes through...Ch. 9.3 - Find the function f such that...Ch. 9.3 - Solve the differential equation y=x+y by making...Ch. 9.3 - Solve the differential equation xy=y+xey/x by...Ch. 9.3 - a Solve the differential equation y=2x1y2. b Solve...Ch. 9.3 - Solve the equation eyy+cosx=0 and graph several...Ch. 9.3 - Solve the initial-value problem y=(sinx)/siny,...Ch. 9.3 - Solve the equation y=xx2+1/(yey) and graph several...Ch. 9.3 - a Use a computer algebra system to draw a...Ch. 9.3 - 2728 a Use a computer algebra system to draw a...Ch. 9.3 - 2932 Find the orthogonal trajectories of the...Ch. 9.3 - 2932 Find the orthogonal trajectories of the...Ch. 9.3 - 2932 Find the orthogonal trajectories of the...Ch. 9.3 - 2932 Find the orthogonal trajectories of the...Ch. 9.3 - 3335 An integral equation is an equation that...Ch. 9.3 - 3335 An integral equation is an equation that...Ch. 9.3 - 3335 An integral equation is an equation that...Ch. 9.3 - Find a function f such that f(3)=2 and...Ch. 9.3 - Solve the initial-value problem in Exercise 9.2.27...Ch. 9.3 - In Exercise 9.2.28 we discussed a differential...Ch. 9.3 - In Exercise 9.1.15 we formulated a model for...Ch. 9.3 - In an elementary chemical reaction, single...Ch. 9.3 - In contrast to the situation of Exercise 40,...Ch. 9.3 - A sphere with radius 1 m has temperature 15C. It...Ch. 9.3 - A glucose solution is administered intravenously...Ch. 9.3 - A certain small country has 10 billion in paper...Ch. 9.3 - A tank contains 1000 L of brine with 15 kg of...Ch. 9.3 - The air in a room with volume 180m3 contains 0.15...Ch. 9.3 - A vat with 500 gallons of beer contains 4 alcohol...Ch. 9.3 - A tank contains 1000 L of pure water. Brine that...Ch. 9.3 - When a raindrop falls, it increases in size and so...Ch. 9.3 - An object of mass m is moving horizontally through...Ch. 9.3 - Allometric growth in biology refers to...Ch. 9.3 - A model for tumor growth is given by the Gompertz...Ch. 9.3 - Let A(t) be the area of a tissue culture at time t...Ch. 9.3 - According to Newtons Law of Universal Gravitation,...Ch. 9.4 - 12 A population grows according to the given...Ch. 9.4 - 1-2 A population grows according to the given...Ch. 9.4 - Suppose that a population develops according to...Ch. 9.4 - Suppose that a population grows according to a...Ch. 9.4 - The Pacific halibut fishery has been modeled by...Ch. 9.4 - Suppose a population P(t) satisfies...Ch. 9.4 - Suppose a population grows according to a logistic...Ch. 9.4 - The table gives the number of yeast cells in a new...Ch. 9.4 - The population of the world was about 6.1 billion...Ch. 9.4 - a Assume that the carrying capacity for the US...Ch. 9.4 - One model for the spread of a rumor is that the...Ch. 9.4 - Biologists stocked a lake with 400 fish and...Ch. 9.4 - a Show that if P satisfies the logistic equation...Ch. 9.4 - For a fixed value of M say M=10, the family of...Ch. 9.4 - The table gives the midyear population of Japan,...Ch. 9.4 - The table gives the midyear population of Japan,...Ch. 9.4 - Consider a population P=P(t) with constant...Ch. 9.4 - Let c be a positive number. A differential...Ch. 9.4 - Lets modify the logistic differential equation of...Ch. 9.4 - Consider the differential equation...Ch. 9.4 - There is considerable evidence to support the...Ch. 9.4 - Another model for a growth function for a limited...Ch. 9.4 - In a seasonal-growth model, a periodic function of...Ch. 9.4 - Suppose we alter the differential equation in...Ch. 9.4 - Graphs of logistic functions Figures 2 and 3 look...Ch. 9.5 - 14 Determine whether the differential equation is...Ch. 9.5 - 14 Determine whether the differential equation is...Ch. 9.5 - 14 Determine whether the differential equation is...Ch. 9.5 - 14 Determine whether the differential equation is...Ch. 9.5 - 514 Solve the differential equation. y+y=1Ch. 9.5 - 514 Solve the differential equation. yy=exCh. 9.5 - 514 Solve the differential equation. y=xyCh. 9.5 - 514 Solve the differential equation....Ch. 9.5 - 514 Solve the differential equation. xy+y=xCh. 9.5 - 514 Solve the differential equation. 2xy+y=2xCh. 9.5 - 514 Solve the differential equation. xy2y=x2,x0Ch. 9.5 - 514 Solve the differential equation. y+2xy=1Ch. 9.5 - 514 Solve the differential equation....Ch. 9.5 - 514 Solve the differential equation....Ch. 9.5 - 1520 Solve the initial-value problem....Ch. 9.5 - 1520 Solve the initial-value problem....Ch. 9.5 - 1520 Solve the initial-value problem....Ch. 9.5 - 1520 Solve the initial-value problem....Ch. 9.5 - 1520 Solve the initial-value problem....Ch. 9.5 - 1520 Solve the initial-value problem....Ch. 9.5 - 2122 Solve the differential equation and use a...Ch. 9.5 - 2122 Solve the differential equation and use a...Ch. 9.5 - A Bernoulli differential equation named after...Ch. 9.5 - 2425 Use the method of Exercise 23 to solve the...Ch. 9.5 - 2425 Use the method of Exercise 23 to solve the...Ch. 9.5 - Solve the second-order equation xy+2y=12x2 by...Ch. 9.5 - In the circuit shown in Figure 4, a battery...Ch. 9.5 - In the circuit shown in Figure 4, a generator...Ch. 9.5 - The figure shows a circuit containing an...Ch. 9.5 - In the circuit of Exercise 29, R=2,C=0.01F,Q(0)=0...Ch. 9.5 - Let P(t) be the performance level of someone...Ch. 9.5 - Two new workers were hired for an assembly line....Ch. 9.5 - In Section 9.3 we looked at mixing problems in...Ch. 9.5 - A tank with a capacity of 400 L is full of a...Ch. 9.5 - An object with mass m is dropped from rest and we...Ch. 9.5 - If we ignore air resistance, we can conclude that...Ch. 9.5 - a Show that the substitution z=1/P transforms the...Ch. 9.5 - To account for seasonal variation in the logistic...Ch. 9.6 - For each predator-prey system, determine which of...Ch. 9.6 - Each system of differential equations is a model...Ch. 9.6 - The system of differential equations...Ch. 9.6 - Lynx eat snowshoe hares and snowshoe hares eat...Ch. 9.6 - 56 A phase trajectory is shown for populations of...Ch. 9.6 - 56 A phase trajectory is shown for populations of...Ch. 9.6 - 78 Graphs of populations of two species are shown....Ch. 9.6 - 78 Graphs of populations of two species are shown....Ch. 9.6 - In Example 1b we showed that the rabbit and wolf...Ch. 9.6 - Populations of aphids and ladybugs are modeled by...Ch. 9.6 - In Example 1 we used Lotka-Volterra equations to...Ch. 9.6 - In Exercise 10 we modeled populations of aphids...Ch. 9.R - a What is a differential equation? b What is the...Ch. 9.R - What can you say about the solutions of the...Ch. 9.R - What is a direction field for the differential...Ch. 9.R - Explain how Eulers method works.Ch. 9.R - What is a separable differential equation? How do...Ch. 9.R - What is a first-order linear differential...Ch. 9.R - a Write a differential equation that expresses the...Ch. 9.R - a Write the logistic differential equation. b...Ch. 9.R - a Write Lotka-Volterra equations to model...Ch. 9.R - Determine whether the statement is true or false....Ch. 9.R - Determine whether the statement is true or false....Ch. 9.R - Determine whether the statement is true or false....Ch. 9.R - Determine whether the statement is true or false....Ch. 9.R - Determine whether the statement is true or false....Ch. 9.R - Determine whether the statement is true or false....Ch. 9.R - Determine whether the statement is true or false....Ch. 9.R - a A direction field for the differential equation...Ch. 9.R - a Sketch a direction field for the differential...Ch. 9.R - a A direction field for the differential equation...Ch. 9.R - a Use Eulers method with step size 0.2 to estimate...Ch. 9.R - 58 Solve the differential equation. y=xesinxycosxCh. 9.R - 58 Solve the differential equation. dxdt=1t+xtxCh. 9.R - 58 Solve the differential equation. 2yey2y=2x+3xCh. 9.R - 58 Solve the differential equation. x2yy=2x3e1/xCh. 9.R - 911 Solve the initial-value problem....Ch. 9.R - 911 Solve the initial-value problem....Ch. 9.R - 911 Solve the initial-value problem....Ch. 9.R - Solve the initial-value problem y=3x2ey,y(0)=1,...Ch. 9.R - 1314 Find the orthogonal trajectories of the...Ch. 9.R - 13-14 Find the orthogonal trajectories of the...Ch. 9.R - a Write the solution of the initial-value problem...Ch. 9.R - a The population of the world was 6.1 billion in...Ch. 9.R - The von Bertalanffy growth model is used to...Ch. 9.R - A tank contains 100 L of pure water. Brine that...Ch. 9.R - One model for the spread of an epidemic is that...Ch. 9.R - The Brentano-Stevens Law in psychology models the...Ch. 9.R - The transport of a substance across a capillary...Ch. 9.R - Populations of birds and insects are modeled by...Ch. 9.R - Suppose the model of Exercise 22 is replaced by...Ch. 9.R - Barbara weighs 60 kg and is on a diet of 1600...Ch. 9.P - Find all functions f such that f is continuous and...Ch. 9.P - A student forgot the Product Rule for...Ch. 9.P - Let f be a function with the property that f(0)=1,...Ch. 9.P - Find all functions f that satisfy the equation...Ch. 9.P - Find the curve y=f(x) such that f(x)0, f(0)=0,...Ch. 9.P - A subtangent is a portion of the x-axis that lies...Ch. 9.P - A peach pie is taken out of the oven at 5:00 PM....Ch. 9.P - Snow began to fall during the morning of February...Ch. 9.P - A dog sees a rabbit running in a straight line...Ch. 9.P - a Suppose that the dog in Problem 9 runs twice as...Ch. 9.P - A planning engineer for a new alum plant must...Ch. 9.P - Find the curve that passes through the point 3, 2...Ch. 9.P - Recall that the normal line to a curve at a point...Ch. 9.P - Find all curves with the property that if the...Ch. 9.P - Find all curves with the property that if a line...Ch. 9.P - a An outfielder fields a baseball 280 ft away from...

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