Prof. A. Suciu Name: depth2pt height-1.6pt width5truecm

MTH 1733 QUIZ 4 Fall 1997

• 6 points Solve the following initial value problem:

  $$y'-3y=3x+5,\quad y(0)=4.$$

  
  

• 6 points Find the general solution of the following system of differential equations:

  $$\begin{array} {@{}r@{\:}c@{\:}r@{\:}c@{\:}r@{\;}c@{\;}l@{}} x'&=&2x&+&y \\ y'&=&-x&+&4y\end{array}$$

  
  

• 6 points A tank initially contains 20 kg of salt dissolved in 200 liters of water. A brine solution containing 3 kg of salt/liter flows into the tank at the rate of 2 liters/minute. The mixture, kept uniform by stirring, flows out at the rate of 4 liters/minute. Set up an appropriate one-compartment model and write down the corresponding initial value problem (do not solve).

  
  
  

• 6 points A certain bacteria culture grows at a rate proportional to its size, doubling every hour. The culture contains 5 million bacteria at time t=0 (with time in hours).
  •   Write down an initial value problem that models the growth of the bacteria culture.
    
  • Solve the intial value problem from part one.
    
  • At what time will there be 200 million bacteria present?
    

• 6 points Consider the following autonomous differential equation:

  y'=y² - 2y -3

  • Find the equilibrium solutions, specifying whether they are stable or unstable.
    
    
  •  Sketch the solution curves corresponding to the initial values y(0)=-2, y(0)=2, y(0)=4.
    
    
    
  • For each of the three curves y=y(t) in part two, decide whether an inflection point occurs, and, if it does, at what value of y.