Dr. Cleary
Math 81
1) Find the general solution to the differential equation:
2) Find the general solution to the following differential
equation:
3) Find the general solution to the following differential
equation:
4) Find the general solution to the following
differential equation
5) Find the particular solution of the given
differential equation subject to the given initial condition:
6) Let 
be the population of bears in Yosemite Valley.
A scientist noted that bears reproduce more when there are tourists
with camcorders nearby. She determined that the rate of change in the
bear population is proportional to the product of the size of the bear
population and the natural logarithm of the ratio of camcorders to
bears. If the number of tourists with camcorders in the park is 
,
write a differential equation satisfied by the bear population.
7) Solve the differential equation (y+1) dx - x dy = 0 by using an integrating factor.
8) Sam wants to make some extra cash on the weekend. He decides to sell lemonade in his neighborhood. He has a 10 gallon cooler for the lemonade. His first batch comes out much too sweet. He decides to dilute the lemonade but the cooler is already full. So he decides to add some plain water to the top at the rate of 1 gallon per minute while he drains the mixture in the cooler at the same rate out of the bottom spigot. Originally, there were 5 pounds of sugar in the 10 gallon cooler. A tasty lemonade mixture has 6 ounces of sugar per gallon. How long until his lemonade is properly mixed, assuming that Sam is stirring his lemonade well?
9) Solve 
.
10) Sketch a few isoclines for the direction field for the
differential equation
and sketch a few solution curves
11) Write down an equation satisfied by the family of circles tangent to the line y=7 with centers on the line x=2. Write down a differential equation satisfied by an orthogonal family of curves to that family.
12) Solve the differential equations 
where
a and b are constants.
13) Newton's law of cooling says that the rate of change
of the temperature of an object is proportional to the difference
between the object's temperature and the surrounding temperture. This
can be useful in establishing the time of death of a corpse. Suppose
that Kermit the Frog is the coroner in Fresno County and that he is
investigating the death of Miss Piggy, who was found dead at the
fountain on campus lying next to her bicycle. At 8am Tuesday morning,
he measures her body temperature at 
. By 9am, her
body has cooled to 
. If the outside
temperature is 
and the normal body temperature
for a pig is 
, when did the helmetless Miss Piggy have
her fatal bike accident?
14) Solve the differential equation :
15) Solve the differential equation:
16) Solve the differential equation:
17) Write down the differential equation for the current in
a circuit which has a battery of voltage 
, a resistor of resistance
R, a capacitor of size C and an inductor with inductance L in that order
in a loop, arranged as in the sketch below:
18) Set up the Picard iteration technique to solve the
initial value problem 
and do the first two
iterations if possible.
19) On what interval is the solution of 
with initial condition 
guaranteed by our 1st
order existence theorem?
20) For what initial conditions can we guarrantee uniqueness of solution for the differential equation of question 19?