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Mathematical Expression Editor

Exercises relating to the application of ODEs to solve problems.

Find the orthogonal trajectories of the given family (represented by the purple
curves in the plot below): Write your answer as , where both and vanish at the
origin. Moreover, you should multiply by constants if necessary so that and .

Find the orthogonal trajectories of the given family (represented by the purple
curves in the plot below): Write your answer as , where

Find the orthogonal trajectories of the given family: Write your answer as , where
both and vanish at the origin. Moreover, you should multiply by constants if
necessary so that and .

It takes about three days to defrost a 10 pound imitation frozen turkey
in a home refrigerator. Suppose that the initial temperature is C and the
ambient temperature is C inside the refrigerator. If after exactly two days, the
temperature of the frozen turkey is C, find a formula for the temperature (in
degrees Celsius) as a function of time for all , where is measured in days

At time , a cup of tea has temperature C. Ten minutes later, its temperature is C.
Ten minutes after that, the temperature is then C. Assuming the temperature obeys
Newton’s Law of cooling, what is the temperature (in degrees Celsius) as
a function of time for all , where is measured in minutes?

Suppose is
the ambient temperature. We know that because every ten minutes, the
difference between tea temperature and room temperature decreases by the same
factor.

A 1000 liter tank is filled with a sugar solution: 200 kilograms of sugar dissolved in
1000 liters of pure water. The solution in the tank is kept thoroughly mixed
at all times. At time , the attendants begin adding 1 liter per minute of
dissolved sugar at a concentration of 0.1 kilograms per liter. At the same time,
One liter per minute is drained from the tank to keep the overall volume of
solution in the tank at 1000 liters. Find a function for the total amount of
sugar in the tank (in kilograms) a time , where is measured in minutes.

A small flask contains 10mL of solvent in which 1 gram of substance X is initially
dissolved. The solution is slowly drained at a rate of 3mL per hour while 4mL per
hour of pure solvent is added (with the whole solution being kept thoroughly mixed).
Find a formula for the amount of substance (in grams) sill in the beaker at time ,
where is measured in hours.

Sample Exam Questions

A tank contains 100 gallons of water in which 300 pounds of salt are dissolved. At
some initial time, workers begin pumping in fresh water, i.e., containing
no salt, at a rate of 10 gallons per minute. During the process, the tank is
kept well-mixed and 20 gallons per minute of the resulting saltwater are
pumped out of the tank (in particular, note that the tank will be empty
after 10 minutes). Find the total amount of salt in the tank (measured in
pounds) which remains 9 minutes after the process starts.