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Spr14-MAT2680-Exam1Review
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MAT 2680 - Spring 2014 - CityTech

Prof Reitz

Exam 1 Review

Sec 1.1-1.3, 2.1, 2.2, 2.4, 2.6

What are you responsible for knowing? In addition to the various methods for solving differential equations, you must know derivative and integration methods including basic functions, u-substitution, integration by parts, and the arctangent integration formula

What will you be given? Any differential equations modeling specific situations such as velocity of falling objects or populations.

  1. The velocity of an object falling near sea level can be modeled by the differential equation .  Suppose a 90kg man is falling near sea level.  His drag coefficient has been estimated as 3kg/s.  Use the approximation  for gravity.
  1. Find the general solution describing the velocity of the man.
  2. Find the equilibrium solution.
  3. Suppose the man fell out of a plane at time t=0.  What was his initial velocity in the downward direction?  Find the particular solution describing his velocity.
  1. The population p(t) of a group of field mice over time can be modeled by the differential equation , where r is the growth rate and k is the predation rate.   Suppose that the growth rate for the mice living in a certain region is 0.7, and the local housecats kill 37 mice per week. Assume a month consists of exactly 4 weeks.
  1. Find the general solution describing the population of mice.
  2. Find the equilibrium solution.
  3. What is the meaning of the equilibrium solution in this situation? Your answer should mention mice, reproduction, and predators.
  4. Suppose there is an initial population of 143 mice.  Find the particular solution describing the population p(t) over time.
  5. In part d. above, what is the eventual result in terms of population? If we come back at some time far in the future, will we find mice there? How many?

For each problem below,  clearly identify the type of equation or solution method: Separable Equation, Homogeneous Equation, Bernoulli Equation, or Linear Equation/Integrating Factor.  Find the general solution (give an explicit solution unless instructed otherwise).  If there is an initial condition given, also find the particular solution satisfying the condition.  

  3. ,  
  6. , solve implicitly

Exam 1 Review ANSWER KEY

If you discover an error please let me know, either in class, on the OpenLab, or by email to jreitz@citytech.cuny.edu.  Corrections will be posted on the “Exam Reviews” page.

  1. a.  
    b.
    c.  His initial velocity is 0, that is     .  
        His velocity is given by:      
  2. a.
    b.
    c. The equilibrium solution gives the population size for which the number of new mice born each month is exactly balanced by the number killed by predators, so the population remains constant.
    d.
    e. The population will decrease over time until no mice remain.
  3. Separable.  
  4. Bernoulli Equation.  
  5. Linear. General solution: .
    Particular solution when     :  
  6. Separable.
  7. Separable.
  8. Homogeneous.
  9. Linear.