Chapter 13 More Differential Equations by azw20493


									Chapter 13

More Differential Equations

Consider the differential equation
                                                  = a − by
where a, b are constants.
 (a) Show that the function
                                               y(t) =  − Ce−bt
       satisfies the above differential equation for any constant C.
 (b) Show that by setting
                                                 C=       − y0
       we also satisfy the initial condition
                                                  y(0) = y0 .
       Remark: You have now shown that the function
                                                          a −bt a
                                            y(t) = y0 −      e +
                                                          b         b
       is a solution to the initial value problem (i.e differential equation plus initial condition)
                                              = a − by,    y(0) = y0 .

For each of the following, show the given function y is a solution to the given differential equation.
 (a) t ·    = 3y, y = 2t3 .
       d2 y
 (b)        + y = 0, y = −2 sin t + 3 cos t.
       d2 y    dy
 (c)        − 2 + y = 6et , y = 3t2 et .
       dt2     dt

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Math 102 Problems                                                                          Chapter 13

Show the function determined by the equation 2x2 +xy −y 2 = C, where C is a constant and 2y = x,
is a solution to the differential equation (x − 2y)    = −4x − y.

Find the constant C that satisfies the given initial conditions.

 (a) 2x2 − 3y 2 = C, y|x=0 = 2.

 (b) y = C1 e5t + C2 te5t , y|t=0 = 1 and      |
                                            dt t=0
                                                     = 0.

 (c) y = C1 cos(t − C2 ), y|t= π = 0 and
                                               | π
                                            dt t= 2
                                                      = 1.

13.5      Friction and terminal velocity
The velocity of a falling object changes due to the acceleration of gravity, but friction has an effect
of slowing down this acceleration. The differential equation satisfied by the velocity v(t) of the
falling object is
                                                = g − kv
where g is acceleration due to gravity and k is a constant that represents the effect of friction. An
object is dropped from rest from a plane.

 (a) Find the function v(t) that represents its velocity over time.

 (b) What happens to the velocity after the object has been falling for a long time (but before it
     has hit the ground)?

13.6      Alcohol level
Alcohol enters the blood stream at a constant rate k gm per unit time during a drinking session.
The liver gradually converts the alcohol to other, non-toxic byproducts. The rate of conversion per
unit time is proportional to the current blood alcohol level, so that the differential equation satisfied
by the blood alcohol level is
                                                = k − sc
where k, s are positive constants. Suppose initially there is no alcohol in the blood. Find the blood
alcohol level c(t) as a function of time from t = 0, when the drinking started.

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Math 102 Problems                                                                            Chapter 13

13.7       Newton’s Law of Cooling
Newton’s Law of Cooling states that the rate of change of the temperature of an object is propor-
tional to the difference between the temperature of the object, T , and the ambient (environmental)
temperature, E. This leads to the differential equation
                                               = k(E − T )
where k > 0 is a constant that represents the material properties and, E is the ambient temperature.
(We will assume that E is also constant.)
 (a) Show that the function
                                          T (t) = E + (T0 − E)e−kt
      which represents the temperature at time t satisfies this equation.
 (b) The time of death of a murder victim can be estimated from the temperature of the body if it is
     discovered early enough after the crime has occurred. Suppose that in a room whose ambient
     temperature is E = 20 degrees C, the temperature of the body upon discovery is T = 30
     degrees, and that a second measurement, one hour later is T = 25 degrees. Determine the
     approximate time of death. (You should use the fact that just prior to death, the temperature
     of the victim was 37 degrees.)

13.8       A cup of coffee
The temperature of a cup of coffee is initially 100 degrees C. Five minutes later, (t = 5) it is 50
degrees C. If the ambient temperature is A = 20 degrees C, determine how long it takes for the
temperature of the coffee to reach 30 degrees C.

13.9       Glucose solution in a tank
A tank that holds 1 liter is initially full of plain water. A concentrated solution of glucose, containing
0.25 gm/cm3 is pumped into the tank continuously, at the rate 10 cm3 /min and the mixture (which
is continuously stirred to keep it uniform) is pumped out at the same rate. How much glucose will
there be in the tank after 30 minutes? After a long time? (Hint: write a differential equation for
c, the concentration of glucose in the tank by considering the rate at which glucose enters and the
rate at which glucose leaves the tank.)

13.10        Pollutant in a lake
(From the Dec 1993 Math 100 Exam) A lake of constant volume V gallons contains Q(t) pounds of
pollutant at time t evenly distributed throughout the lake. Water containing a concentration of k
pounds per gallon of pollutant enters the lake at a rate of r gallons per minute, and the well-mixed
solution leaves at the same rate.
 (a) Set up a differential equation that describes the way that the amount of pollutant in the lake
     will change.

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Math 102 Problems                                                                          Chapter 13

 (b) Determine what happens to the pollutant level after a long time if this process continues.

 (c) If k = 0 find the time T for the amount of pollutant to be reduced to one half of its initial

A barrel initially contains 2 kg of salt dissolved in 20 L of water. If water flows in the rate of 0.4 L
per minute and the well-mixed salt water solution flows out at the same rate. How much salt is
present after 8 minutes?

13.12            Slope fields
Consider the differential equations given below. In each case, draw a slope field, determine the
values of y for which no change takes place [such values are called steady states] and use your slope
field to predict what would happen starting from an initial value y(0) = 1.
 (a)       = −0.5y
 (b)       = 0.5y(2 − y)
 (c)       = y(2 − y)(3 − y)

Draw a slope field for each of the given differential equations:
 (a)    dt
             = 2 + 3y
 (b)    dt
             = −y(2 − y)
 (c)    dt
             = 2 − 3y + y 2
 (d)    dt
             = −2(3 − y)2
 (e)    dt
             = y2 − y + 1
  (f)   dt
             = y3 − y
        dy       √
 (g)    dt
             =       y(y − 2)(y − 3)2 , y ≥ 0.
 (h)    dt
             = 2ey − 2
  (i)   dt
             = A − sin y (Hint: consider the cases A < −1, A = −1, −1 < A < 1, A = 1 and A > 1).
  (j)   dt
             − y = et

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Math 102 Problems                                                                       Chapter 13

For each of the differential equations (a) to (i) in Problem 13.13, plot dy as a function of y, draw
the motion along the y axis, identify the steady state(s) and indicate if the motions are toward or
away from the steady state(s).

13.15       Periodic motion
 (a) Show that the function y(t) = A cos(wt) satisfies the differential equation

                                              d2 y
                                                   = −w 2 y

     where w > 0 is a constant, and A is an arbitrary constant. [Remark: Note that w corresponds
     to the frequency and A to the amplitude of an oscillation represented by the cosine function.]

 (b) It can be shown using Newton’s Laws of motion that the motion of a pendulum is governed
     by a differential equation of the form

                                            d2 y    g
                                                 = − sin(y),
                                            dt      L

     where L is the length of the string, g is the acceleration due to gravity (both positive con-
     stants), and y(t) is displacement of the pendulum from the vertical. What property of the
     sine function is used when this equation is approximated by the Linear Pendulum Equation:

                                              d2 y    g
                                                   = − y.
                                              dt      L

 (c) Based on this Linear Pendulum Equation, what function would represent the oscillations?
     What would be the frequency of the oscillations?

 (d) What happens to the frequency of the oscillations if the length of the string is doubled?

13.16       A sugar solution
Sugar dissolves in water at a rate proportional to the amount of sugar not yet in solution. Let Q(t)
be the amount of sugar undissolved at time t. The initial amount is 100 kg and after 4 hours the
amount undissolved is 70 kg.

 (a) Find a differential equation for Q(t) and solve it.

 (b) How long will it take for 50 kg to dissolve?

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Math 102 Problems                                                                          Chapter 13

13.17       Infant weight gain
During the first year of its life, the weight of a baby is given by
                                           y(t) = 3t + 64

where t is measured in some convenient unit.

 (a) Show that y satisfies the differential equation

                                                  dy   k
                                                  dt   y

     where k is some positive constant.

 (b) What is the value for k?

 (c) Suppose we adopt this differential equation as a model for human growth. State concisely (that
     is, in one sentence) one feature about this differential equation which makes it a reasonable
     model. State one feature which makes it unreasonable.

13.18       Cubical crystal
A crystal grows inside a medium in a cubical shape with side length x and volume V. The rate of
change of the volume is given by
                                          = kx2 (V0 − V )
where k and V0 are positive constants.
 (a) Rewrite this as a differential equation for   dt

 (b) Suppose that the crystal grows from a very small “seed.” Show that its growth rate continually

 (c) What happens to the size of the crystal after a very long time?

 (d) What is its size (that is, what is either x or V ) when it is growing at half its initial rate?

13.19       Leaking water tank
A cylindrical tank with cross-sectional area A has a small hole through which water drains. The
height of the water in the tank y(t) at time t is given by:

                                                 √     kt 2
                                         y(t) = ( y0 −    )
where k, y0 are constants.

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Math 102 Problems                                                                          Chapter 13

 (a) Show that the height of the water, y(t), satisfies the differential equation

                                               dy    k√
                                                  =−    y.
                                               dt    A

 (b) What is the initial height of the water in the tank at time t = 0 ?

 (c) At what time will the tank be empty ?

 (d) At what rate is the volume of the water in the tank changing when t = 0?

Find those constants a, b so that y = ex and y = e−x are both solutions of the differential equation

                                           y + ay + by = 0.

Let y = f (t) = e−t sin t, − ∞ < t < ∞.

 (a) Show that y satisfies the differential equation y + 2y + 2y = 0.

 (b) Find all critical points of f (t).

A biochemical reaction in which a substance S is both produced and consumed is investigated. The
concentration c(t) of S changes during the reaction, and is seen to follow the differential equation

                                          dc         c
                                             = Kmax     − rc
                                          dt        k+c
where Kmax , k, r are positive constants with certain convenient units. The first term is a concentration-
dependent production term and the second term represents consumption of the substance.

 (a) What is the maximal rate at which the substance is produced? At what concentration is the
     production rate 50% of this maximal value?

 (b) If the production is turned off, the substance will decay. How long would it take for the
     concentration to drop by 50%?


At what concentration does the production rate just balance the consumption rate?

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