MTH 142 Practice Problems for Exam 3
This is a selection of sample problems from sections 10.1 through 10.7, and applications F,G.

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  1. 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).

  2.  
    (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 $\frac{dy}{dx} = 2.5 y - 7.5$.

  5. Find an equilibrium solution and determine from the plot if it is stable or unstable. Explain your answer.
  6. Consider the differential equation dy/dt  = y2 - k2  where k is a positive constant.  What are the equilibrium solutions? Which one of the equilibrium solutions is approached by a solution that satisfies y=0 when t=0 ? Explain.
  7. A yam has been heated to 170 degrees Fahrenheit and is placed in a room whose temperature is maintained at 70 degrees F. In half an hour the yam cools to 100 degrees F. Let Y(t) be the temperature of the yam at time t hours after it was placed in the room.

  8. a) Assume the Newton's law of cooling applies, write a differential equation satisfied by Y(t).
    b) Solve the differential equation to find a formula for Y(t).
  9. A drug is given intravenously at a rate of 5 mg/hr. The drug is excreted frmo the patient's body at a rate proportional to the amount present, with constant of proportionality equal to 0.2. No drug is present at time 0.

  10. a) Write a differential equation satisfied by Q(t), the amount of drug in the patient's body at time t..
    b) If treatment is continued for a very long time, how much of the drug will be present in the body?
  11. Solve the initial value problems:

  12. 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 = 2 y2 ,           y=1 when x = 0.
    d)  y' + y' sin ( y ) = x / y,        y=0 when x = 0.
  13. The growth of certain population is described by the differential equation

  14. $\frac{dP}{dt} = 0.001 (800 P - P^2)$,
    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.
  15. An object with mass 2 Kg is thrown up with initial velocity of 30 m/s.

  16. a) If the gravitational force is the only force acting on the object, obtain a differential equation satisfied by the velocity v(t). Explain.
    b) If in addition to the gravitational force, there is an air friction force that is proportional to the magnitude of the velocity, what is the resulting differential equation satisfied by v(t)? Explain.
    c) Under the conditions of part (b), what is a formula for v(t)? (give the answer in terms of the proportionality constant).
  17. A radioactive element decomposes so that at any given time, the rate of change of mass is proportional to the mass present.

  18. 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?
  19. Consider the motion of a particle in the XY plane given by the parametric equations

  20. x = 4t - t2 , y = t - 5, for t >= 0.
    a) Is the particle ever in the second quadrant? When?
    b) What is the speed of the particle when t = 4?
    c) What is the distance traveled between t=0 and t=4? (Use Simpson's rule to evaluate the integral).
  21. The following is data corresponding to certain population.
  22. 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 $P$.
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