Chapter 9

Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. \( y = \frac {1}{x + k} \)

In Section 9.3 we looked at mixing problems in which the volume of fluid remained constant and saw that such problems give rise to separable differentiable equations. (See Example 6 in that section.) If the rates of flow into and out of the system are different, then the volume is not constant and the resulting differential equation is linear but not separable. A tank contains 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5 L/min. The solution is kept mixed and is drained from the tank at a rate of 3 L/min. If \( y(t) \) is the amount of salt (in kilograms) after \( t \) minutes, show that \( y \) satisfies the differential equation \( \frac {dy}{dt} = 2 - \frac {3y}{100 + 2t} \) Solve this equation and find the concentration after 20 minutes.

An integral equation is an equation that contains an unknown function \( y(x) \) and an integral that involves \( y(x). \) Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.] \( y(x) = 2 + \int^x_2 [t - ty(t)] dt \)

A tank with a capacity of 400 L is full of a mixture of water and chlorine with a concentration of 0.05 g of chlorine per liter. In order to reduce the concentration of chlorine, fresh water is pumped into the tank at a rate of 4 L/s. The mixture is kept stirred and is pumped out at a rate of 10 L/s. Find the amount of chlorine in the tank as a function of time.

An integral equation is an equation that contains an unknown function \( y(x) \) and an integral that involves \( y(x). \) Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.] \( y(x) = 2 + \int^x_1 \frac {dt}{ty(t)}, x > 0 \)

An object with mass \( m \) is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If \( s(t) \) is the distance dropped after \( t \) seconds, then the speed is \( v = s'(t) \) and the acceleration is \( a = v'(t). \) If \( g \) is the acceleration due to gravity, them the downward force on the object is \( mg - cv, \) where \( c \) is a positive constant, and Newton's Second Law gives \( m \frac {dv}{dt} = mg - cv \) (a) Solve this as a linear equation to show that \( v = \frac {mg}{c} (1 - e^{-ct/m}) \) (b) What is the limiting velocity? (c) Find the distance the object has fallen after \( t \) seconds.

An integral equation is an equation that contains an unknown function \( y(x) \) and an integral that involves \( y(x). \) Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.] \( y(x) = 4 + \int^x_0 2t \sqrt {y(t)} dt \)

Find a function \( f \) such that \( f(3) = 2 \) and \( (t^2 + 1)f'(t) + [f(t)]^2 + 1 = 0 t \neq 1 \)

(a) Show that the substitution \( z = 1/P \) transforms the logistic differential equation \( P' = kP(1 - P/M) \) into the linear differential equation \( z' + kz = \frac {k}{M} \) (b) Solve the linear differential equation in part (a) and thus obtain an expression for \( P(t). \) Compare with Equation 9.4.7.

To account for seasonal variation in the logistic differential equation, we could allow \( k \) and \( M \) to be functions of \( t: \) \( \frac {dP}{dt} = k(t)P (1 - \frac {P}{M(t)}) \) (a) Verify that the substitution \( z = 1/P \) transform this equation into the linear equation \( \frac {dz}{dt} + k(t)z = \frac {k(t)}{M(t)} \) (b) Write an expression for the solution of the equation in part (a) and use it to show that if the carrying capacity \( M \) is constant, then \( P(t) = \frac {M}{1 + CMe^{-\int k(t) dt}} \) Deduce that if \( \int^\infty_0 k(t) dt = \infty, \) then \( \lim_{t \to \infty} P(t) = M. \) [This will be true if \( k(t) = k_0 + a \cos bt \) with \( k_0 > 0, \) which describes a positive instrinsic growth rate with a periodic seasonal variation.] (c) If \( k \) is constant but \( M \) varies, show that \( z(t) = e^{-kt} \int^t_0 \frac {ke^{ks}}{M(s)} ds + Ce^{-kt} \) and use 1'Hospital's Rule to deduce that if \( M(t) \) has a limit as \( t \to \infty, \) then \( P(t) \) has the same limit.

In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product \( C: A + B \to C. \) The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B: \( \frac {d[C]}{dt} = k [A][B] \) (See Examples 3.7.4.) Thus, if the initial concentrations are \( [A] = a \) moles/L and \( [B] = b \) moles/L and we write \( x = [C], \) then we have \( \frac {dx}{dt} = k(a - x)(b - x) \) (a) Assuming that \( a \neq b, \) find \( x \) as a function of \( t. \) Use the fact that the initial concentration of C is 0. (b) Find \( x(t) \) assuming that \( a = b. \) How does this expression for \( x(t) \) simplify if it is known that \( [C] = \frac {1}{2}a \) after 20 seconds?

A sphere with radius 1 m has temperature \( 15^oC. \) It lies inside a concentric sphere with radius 2 m and temperature \( 25^oC. \) The temperature \( T(r) \) at a distance \( r \) from the common center of the spheres satisfies the differential equation \( \frac {d^2T}{dr^2} + \frac {2}{r}\frac {dT}{dr} = 0 \) If we let \( S = dT/dr, \) then \( S \) satisfies a first-order differential equation. Solve it to find an expression for the temperature \( T(r) \) between the spheres.

A glucose solution is administered intravenously into the bloodstream at a constant rate \( r. \) 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 \( C = C(t) \) of the glucose solution in the bloodstream is \( \frac {dC}{dt} = r - kC \) where \( k \) is a positive constant. (a) Suppose that the concentration at time \( t = 0 \) is \( C_o. \) Determine the concentration at any time \( t \) by solving the differential equation. (b) Assuming that \( C_o < r/k, \) find lim \( _{t \to \infty} C(t) \) and interpret your answer.

A certain small country has 10 billion dollars in paper currency in circulation, and each day 50 million dollars comes into the country's 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 \( x = x(t) \) denote the amount of new currency in circulation at time \( t, \) with \( x(0) = 0. \) (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 \( 90% \) 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 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank (a) after \( t \) minutes and (b) after 20 minutes?

The air in a room with volume 180 \( m^3 \) contains \( 0.15% \) carbon dioxide initially. Fresher air with only \( 0.05% \) carbon dioxide flows into the room at a rate of 2 \( m^3/min \) 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 5 gal/min 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 5L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15L/min. How much salt is in the tank (a) after \( t \) minutes and (b) after one hour?

When a raindrop falls, it increase in size and so its mass at time \( t \) is a function of \( t, \) namely, \( m(t). \) The rate of growth of the mass is \( km(t) \) for some positive constant \( k. \) When we apply Newton's Law of Motion to the raindrop, we get \( (mv)' = gm, \) where \( v \) is the velocity of the raindrop (directed downward) and \( g \) is the acceleration due to gravity. The terminal velocity of the raindrop is lim \( _{t \to \infty} v(t). \) Find an expression for the terminal velocity in terms of \( g \) and \( k. \)

An object of mass \( m \) is moving horizontally through a medium which resists the motion with a force that is a function of the velocity; that is, \( m \frac {d^2s}{dt^2} = m \frac {dv}{dt} = f(v) \) where \( v = v(t) \) and \( s = s(t) \) represent the velocity and position of the object at time \( t, \) respectively. For example, think of a boat moving through the water. (a) Suppose that the resisting force is proportional to the velocity, that is \( f(v) = -kv, k \) a positive constant. (This model is appropriate for small values of \( v. \)) Let \( v(0) = v_0 \) and \( s \) at any time \( t. \) What is the total distance that the object travels from time \( t = 0? \) (b) For larger values of \( v \) a better model is obtained by supposing that the resisting force is proportional to the square of the velocity, that is, \( f(v) = kv^2, k > 0\. \) (This model was first proposed by Newton.) Let \( v_0 \) and \( s_0 \) be the initial values of \( v \) and \( s. \) Determine \( v \) and \( s \) at any time \( t. \) 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 \( L_1(t) \) and \( L_2(t) \) are the sizes of two organs in an organism of age \( t, \) then \( L_1 \) and \( L_2 \) satisfy an allometric law if there specific growth rates are proportional: \( \frac {1}{L_1} \frac {dL_1}{dt} = k \frac {1}{L_2} \frac {dL_2}{dt} \) where \( k \) is a constant. (a) Use the allometric law to write a differential equation relating \( L_1 \) and \( L_2 \) and solve it to express \( L_1 \) as a function of \( L_2. \) (b) In a study of several species of unicellular algae, the proportionality constant in the allometric law relating \( B \) (cell biomass) and \( V \) (cell volume) was found to be \( k = 0.0794. \) Write \( B \) as a function of \( V. \)

A model for tumor growth is given by the Gompertz equation \( \frac {dV}{dt} = a (\ln b - \ln V) V \) where \( a \) and \( b \) are positive constant and \( V \) is the volume of the tumor measured in \( mm^3. \) (a) Find a family of solution for tumor volume as a function of time. (b) Find the solution that has an initial tumor volume of \( V(0) = 1 mm^3. \)

Let \( A(t) \) be the area of a tissue culture at time \( t \) and let \( M \) be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery is proportional to \( \sqrt {A(t)}. \) 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 \( \sqrt {A(t)} \) and \( M - A(t). \) (a) Formulate a differential equation and use it to show that the tissue grows fastest when \( A(t) = \frac {1}{3} M. \) (b) Solve the differential equation to find an expression for \( A(t). \) Use a computer algebra system to perform the integration.