MTH 142 Practice Problems for Exam 3
This is a selection of sample problems from sections 10.1 through
10.6
up to page 526
Consider the differential equation y' =
- x - y, where
is a positive constant. One (and only one) of the following slope field
plots is definitely not associated with this differential equation. Which
plot does not correspond to the differential equation? Explain why
not. Assume that the lower left corner corresponds to (0,0).
(a)
(b)
Suppose that y(x) is a solution of the initial value problem y(0.2)
= 1, y' = x^{2} + y , . Use Euler's method with
to estimate .
Do all the steps by hand.
Make a sketch of the slope field for the differential equation .
Find an equilibrium solution and determine from the plot if it is stable
or unstable. Explain your answer.
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), the amount
of drug in the patient's body at time t..
b) If treatment is continued for a very long time, how much of the
drug will be present in the body?
Solve the initial value problems:
a) , .
b) dr/dx + e^{r} = x e^{r}
, r = 0 when x = 1. c) y' + 8 = 2 y^{2} ,
y=1 when x = 0. d) y' + y' sin ( y ) = x / y,
y=0 when x = 0.
An object with mass 2 Kg is thrown up with initial velocity of 30 m/s.
a) If the gravitational force is the only force acting on the object,
obtain a differential equation satisfied by the velocity v(t). Explain.
b) If in addition to the gravitational force, there is an air friction
force that is proportional to the magnitude of the velocity, what is the
resulting differential equation satisfied by v(t)? Explain.
c) Under the conditions of part (b), what is a formula for v(t)?
(give the answer in terms of the proportionality constant).
A radioactive element decomposes so that at any given time, the rate of
change of mass is proportional to the mass present.
a) Write down a differential equation satisfied by M ( t ),
the mass at time t. Solve the differential equation.
b) It takes 10 years for the mass to reduce to 90 % of the original
amount. If currently there are 3 Kg , how long ago was the mass 5
Kg?
Determine if y = ( c^{2} - x^{2} )^{1/2}
is a solution to the differential equation y y ' = - x .
If the asnwer is yes, specify the interval in which it is a solution.