MTH 1733 -- Honors Calculus 1 -- Fall 1997

Quiz 4

(6 points) Solve the following initial value problem:
$\begin{displaymath} y'-3y=3x+5,\quad y(0)=4.\end{displaymath}$
(6 points) Find the general solution of the following system of differential equations:
$\begin{displaymath} \begin{array} {@{}r@{\:}c@{\:}r@{\:}c@{\:}r@{\;}c@{\;}l@{}} x'&=&2x&+&y \\ y'&=&-x&+&4y\end{array}\end{displaymath}$
(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.

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Created by Alexandru Suciu, Mon Nov 24, 1997

http://www.math.neu.edu/~suciu/mth1733/1733f97.q4/index.html