Answer: A sphere with radius 1 m has temperature . It lies inside a concentric sphere

110 Solve the differential equation.dy dx xy 2

110 Solve the differential equation.dy dx xey x

110 Solve the differential equation.xy 2 y x  1 y

110 Solve the differential equation.y 2  xy 2 y 1 y

110 Solve the differential equation. y  sin yy x  x 3 dv

110 Solve the differential equation.dv ds s  1 sv  s

110 Solve the differential equation.dy dt t yeyt 2

110 Solve the differential equation.dy d ey sin2  y sec 

110 Solve the differential equation.dp dt t 2 p p  t 2 1 dz

110 Solve the differential equation.dz dt  e tz 0

1118 Find the solution of the differential equation that satisfies the given initial condition.dy dx x y y0 3 dy

1118 Find the solution of the differential equation that satisfies the given initial condition.dy dx ln x xy y1 2 du

1118 Find the solution of the differential equation that satisfies the given initial condition.du dt 2t  sec2 t 2u u0 5 y x

1118 Find the solution of the differential equation that satisfies the given initial condition.y xy sin x y  1 y0 1 x ln

1118 Find the solution of the differential equation that satisfies the given initial condition.x ln x y(1  s3  y 2 )y y1 1 dP d

1118 Find the solution of the differential equation that satisfies the given initial condition.dP dt sPt P1 2 yta

1118 Find the solution of the differential equation that satisfies the given initial condition.ytan x a  y y3 a 0  x  2 dL d

1118 Find the solution of the differential equation that satisfies the given initial condition.dL dt kL2 ln t L1 1 0, 1

Find an equation of the curve that passes through the point and whose slope at is .

Find the function such that and

Solve the differential equation by making the change of variable

. Solve the differential equation by making the change of variable

(a) Solve the differential equation . ; (b) Solve the initial-value problem , , and graph the solution. (c) Does the initial-value problem , , have a solution? Explain.

Solve the equation and graph several members of the family of solutions. How does the solution curve change as the constant varies?

Solve the initial-value problem , , and graph the solution (if your CAS does implicit plots).

Solve the equation and graph several members of the family of solutions (if your CAS does implicit plots). How does the solution curve change as the constant varies?

2728 (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 solu tions obtained in part (b). Compare with the curves from part (a). y y 2 y

2728 (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 solu tions obtained in part (b). Compare with the curves from part (a).y xy x

2932 Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.x 2  2y 2 k 2 y

2932 Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.y 2 kx 3

2932 Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.y k x y

2932 Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.y x 1  kx

3335 An integral equation is an equation that contains an unknown function and an integral that involves . Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.]yx 2  y x 2 t tyt dt yx 2

3335 An integral equation is an equation that contains an unknown function and an integral that involves . Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.]yx 2  y x 1 dt ty t x  0 yx

3335 An integral equation is an equation that contains an unknown function and an integral that involves . Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.]yx 4  y x 0 2tsy t dt 9.3

Find a function such that and [Hint: Use the addition formula for on Reference Page 2.]

Solve the initial-value problem in Exercise 27 in Section 9.2 to find an expression for the charge at time . Find the limiting value of the charge.

In Exercise 28 in Section 9.2 we discussed a differential equation that models the temperature of a cup of coffee in a room. Solve the differential equation to find an expression for the temperature of the coffee at time .

In Exercise 15 in Section 9.1 we formulated a model for learning in the form of the differential equation where measures the performance of someone learning a skill after a training time , is the maximum level of per formance, and is a positive constant. Solve this differential equation to find an expression for . What is the limit of this expression?

In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C: . The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B: (See Example 4 in Section 3.7.) Thus, if the initial concentrations are A molesL and B molesL and we write C , then we have (a) Assuming that , find as a function of . Use the fact that the initial concentration of C is 0. (b) Find assuming that . How does this expres sion for simplify if it is known that after 20 seconds?

In contrast to the situation of Exercise 40, experiments show that the reaction satisfies the rate law and so for this reaction the differential equation becomes where and and are the initial concentrations of hydrogen and bromine. (a) Find as a function of in the case where . Use the fact that .(b) If , find as a function of . Hint: In performing the integration, make the substitution

A sphere with radius 1 m has temperature . It lies inside a concentric sphere with radius 2 m and temperature . The temperature at a distance from the common center of the spheres satisfies the differential equation If we let , then satisfies a first-order differential equation. Solve it to find an expression for the temperature between the spheres.

A glucose solution is administered intravenously into the bloodstream at a constant rate . As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Thus a model for the concentration of the glucose solution in the bloodstream is where is a positive constant. (a) Suppose that the concentration at time is . Determine the concentration at any time by solving the differential equation. (b) Assuming that , find and interpret your answer.

A certain small country has $10 billion in paper currency in circulation, and each day $50 million comes into the countrys banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let denote the amount of new currency in circulation at time , with . (a) Formulate a mathematical model in the form of an initial-value problem that represents the flow of the new currency into circulation. (b) Solve the initial-value problem found in part (a). (c) How long will it take for the new bills to account for of the currency in circulation?

A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 Lmin. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank (a) after minutes and (b) after 20 minutes?

The air in a room with volume contains carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of and the mixed air flows out at the same rate. Find the percentage of carbon dioxide in the room as a function of time. What happens in the long run?

A vat with 500 gallons of beer contains 4% alcohol (by volume). Beer with 6% alcohol is pumped into the vat at a rate of and the mixture is pumped out at the same rate. What is the percentage of alcohol after an hour?

A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 Lmin. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 Lmin. The solution is kept thoroughly mixed and drains from the tank at a rate of 15 Lmin. How much salt is in the tank (a) after minutes and (b) after one hour?

When a raindrop falls, it increases in size and so its mass at time is a function of , namely . The rate of growth of the mass is for some positive constant . When we apply New tons Law of Motion to the raindrop, we get , where is the velocity of the raindrop (directed downward) and is the acceleration due to gravity. The terminal velocity of the raindrop is . Find an expression for the terminal velocity in terms of and .

An object of mass is moving horizontally through a medium which resists the motion with a force that is a function of the velocity; that is, where and represent the velocity and position of the object at time , respectively. For example, think of a boat moving through the water. (a) Suppose that the resisting force is proportional to the velocity, that is, , a positive constant. (This model is appropriate for small values of .) Let and be the initial values of and . Determine and at any time . What is the total distance that the object travels from time ? (b) For larger values of a better model is obtained by supposing that the resisting force is proportional to the square of the velocity, that is, , . (This model was first proposed by Newton.) Let and be the initial values of and . Determine and at any time . What is the total distance that the object travels in this case?

Allometric growth in biology refers to relationships between sizes of parts of an organism (skull length and body length, for instance). If and are the sizes of two organs in an organism of age , then and satisfy an allometric law if their specific growth rates are proportional: where is a constant. (a) Use the allometric law to write a differential equation relating and and solve it to express as a function of . (b) In a study of several species of unicellular algae, the proportionality constant in the allometric law relating (cell biomass) and (cell volume) was found to be . Write as a function of .

Homeostasis refers to a state in which the nutrient content of a consumer is independent of the nutrient content of its food. In the absence of homeostasis, a model proposed by Sterner and Elser is given by where and represent the nutrient content of the food and the consumer, respectively, and is a constant with . (a) Solve the differential equation. (b) What happens when ? What happens when ?

Let be the area of a tissue culture at time and let be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery of the tissue and the number of cells on the periphery is proportional to . So a reasonable model for the growth of tissue is obtained by assuming that the rate of growth of the area is jointly proportional to and . (a) Formulate a differential equation and use it to show that the tissue grows fastest when . (b) Solve the differential equation to find an expression for . Use a computer algebra system to perform the integration.

According to Newtons Law of Universal Gravitation, the gravitational force on an object of mass that has been projected vertically upward from the earths surface is where is the objects distance above the surface at time , is the earths radius, and is the acceleration due to gravity. Also, by Newtons Second Law, and so (a) Suppose a rocket is fired vertically upward with an initial velocity . Let be the maximum height above the surface reached by the object. Show that [Hint: By the Chain Rule, .] (b) Calculate . This limit is called the escape velocity for the earth. (c) Use mi and fts to calculate in feet per second and in miles per second.