MTH 142 Practice Problems for Exam 3
This is a selection of sample problems from sections 10.5, and 11.1 through 11.7.

Last updated December 1, 2002

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  1. Find all values of the constant$a$ for which f(x) = $a$x2 is a solution to the differential equation y ' = 3 x + y/x , or show that there are none.
  2. Consider the differential equation y' = $a$ - x - y, where $a$ is a positive constant. One (and only one) of the following slope field plots is definitely not associated with this differential equation. Which plot does not correspond to the differential equation? Explain why not.  Assume that the lower left corner corresponds to (0,0).
     
    (a)
     (b)
  3. Suppose that y(x) is a solution of the initial value problem y(0.2) = 1, y' = x2 + y , . Use Euler's method with $\Delta x = 0.1$ to estimate y(0.5). Do all the steps by hand.
  4. Make a sketch of the slope field for the differential equation
     $\frac{dy}{dx} = 2.5 y - 7.5$.
    Find an equilibrium solution and determine from the plot if it is stable or unstable. Explain your answer.
  5. The number N of certain kind of bacteria in a culture changes at a rate proportional to N. Initially there were 5 million bacteria, and 2 hours later there were 7 million.
    a) Write down a differential equation satisfied by N.
    b) Obtain a formula for N
  6. A piece of meat is taken out of the freezer and placed on the countertop. The temperature of the room is 22 degrees Celsius. The initial temperature of the meat was -5 degrees Celsius, and 3 hours later the temperature of the meat is 10 degrees Celsius. Suppose that the temperature T of the meat at time t satisfies Newton's Law of Heating and Cooling.
    a) Write down a differential equation satisfied by T.
    b) Find T for any time t.
    c) When will the temperature of the meat be equal to 20 degrees Celsius?
  7. A large box is sealed except for two pipes for incoming and outgoing air. The box has volume 10 cubic meters, and it is initially filled with CO2. Pure oxygen enters the box at a constant rate of 2 cubic meters per minute. The oxygen mixes with the air in the box, and the mixture leaves at 2 cubic meters per minute.
    Let Q be the amount of oxygen at time t in the box.
    a) write down a differential equation for Q.
    b) Find Q at any time t
    c) When will be box contain equal parts of oxygen and CO2?
  8. Solve the initial value problems:
    a) $y \frac{dy}{dx} - \frac{x}{y} = 0$$y(1) = 2$.
    b) dr/dx  + er = x er    , r = 0 when x = 1.
    c)  y' + 8 = y ,           y=9 when x = 0.
    d)  y y' + y y' sin ( y ) = x ,        y=0 when x = 0.
  9. The growth of certain population is described by the differential equation
    P' = 0.8 P - 0.001 P2
    where P(t)  is the population at time t days. The initial population is 25 mill.
    a) What is the population at the time when the population is growing most rapidly?
    b) Will the population ever exceed 900 mill.? Explain.
  10. A radioactive element decomposes so that at any given time, the rate of change of mass is proportional to the mass present.
    a) Write down a differential equation satisfied by M ( t ), the mass at time t.  Solve the differential equation.
    b) It takes 10 years for the mass to reduce to 90 % of the original amount.  If currently there are 3 Kg , how long ago was the mass 5 Kg?
  11. The following is data corresponding to certain population.
    t (days)     0      1      2      3      4      5      6 
    P (mill) 3.975 7.2 12.825 23.55 37.65 57.0 79.27
    $\frac{dP}{dt}$              
    $\frac{1}{P} \frac{dP}{dt}$              
    a) Use the data to calculate approximately the blanks in the table, and  plot $\frac{1}{P} \frac{dP}{dt}$ versus $P$,
    b) fit a line as best you can through the points, and then obtain a differential equation satisfied by $P$.
  12. The number of individuals y (in millions) of certain population at time t years satisfies the logistic equation
    y' = 20 y - 2 y2.
    a) What is the carrying capacity of the population?
    b) If the initial population is 5 million, at what rate is the population changing initially?
    c) For what values of the population is the population changing at a rate of 22 mill/year?
  13. Obtain the Fourier polynomial of order 2 of the 2 Pi periodic function which in the interval -Pi < x < Pi is given by f(x) = -x
  14. Obtain the Fourier polynomial of order 1 of the 2 Pi periodic function which in the interval -Pi < x < Pi is given by
    f(x) = -0.5,     if -Pi < x < 0,   and,
    f(x) = 2.0  if 0 < x < Pi

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