- Consider the differential equation yó = 2 - x - y,
where a is a positive constant.
One (and only one) of the following slope field plots
is definitely not associated with this
Which plot does not
correspond to the differential equation?
Explain why not.
- Suppose that y(x) is a solution of the initial value problem
y(0.2) = 1, yó = x2 + y.
Use Euler's method with
Dx = 0.1 to estimate y(0.5).
Make a rough sketch of the slope field for the differential
equation [dy/ dx] = 2.5 y - 7.5.
Find an equilibrium solution and determine from the plot if it is
stable or unstable. Explain why.
- A yam has been heated to 170 degrees Fahrenheit and is placed in a room
whose temperature is maintained at 70 degrees F.
In half an hour the yam cools to 100 degrees F.
Let Y(t) be the temperature of the yam at time t hours
after it was placed in the room.
a) Assume the Newton's law of cooling applies, write a differential equation
satisfied by Y(t).
b) Solve the differential equation to find a formula for Y(t).
- A drug is given intravenously at a rate of 5 mg/hr.
The drug is excreted frmo the patient's body at a rate proportional
to the amount present, with constant of proportionality equal to 0.2.
No drug is present at time 0.
a) Write a differential equation satisfied by Q(t).
b) If treatment is continued for a long, long time,
how much of the drug will be present in the body?
- Solve the initial value problems:
a) y [dy/ dx] - x/y = 0, y(1) = 2.
b) [dr/ dx] + x er = er , r(1) = 0.
- The growth of certain population is
described by the differential equation
[dP/ dt] = 0.001 (800 P - P2),
where P(t) is the population at time t days.
The initial population is 25 mill.
a) What is the population at the time when the population
is growing most rapidly?
b) Will the population ever exceed 900 mill.? Explain.
- The following is data corresponding to certain population.
|t (days) || 0 || 1 || 2 || 3 || 4 || 5 || 6
|P (mill) ||3.975 ||7.2 ||12.825 ||23.55 ||37.65 ||57.0 ||79.27
|[dP/ dt] ||
|1/P [dP/ dt] |
a) Use the data to calculate approximately the blanks in the table.
b) Use (a) to plot 1/P [dP/ dt] versus P,
c) fit a line as best you can through the points, and then
obtain a differential equation satisfied by P.