MATH 308-301,302
Summer 2008
Practice Test I
1. A large tank initially contains 10 L of fresh water. A brine containing 20 g/L of salt flows
into the tank at a rate of 3 L/min. The solution inside the tank is kept well stirred and
flows out of the tank at the rate 2 L/min. Determine the concentration of salt in the tank
as a function of time.
2. A 400-lb object is released from rest 500 ft above the ground and allowed to fall under
the influence of gravity. Assume that the force in pounds due to air resistance is −10v,
where v is the velocity of the object in ft/sec. Determine the equation of motion of the
object.
3. An object with temperature 1500 is placed in a freezer whose temperature is 300 . Assume
that the temperature of the freezer remains essentially constant.
(a) If the object is cooled to 1200 after 8 min, what will its temperature be after 18 min?
(b) When will its temperature be 600 ?
4. Draw the isoclines with their direction markers and sketch the solution to the initial value
problem
yy ′ = −x, y(0) = 4.
5. Determine (without solving the problem) an interval in which the solution to the initial
value problem
y ′ + y tan x = sec x, y(0) = 3
is certain to exist.
6. Solve the following initial value problem
√
ydx + (1 + x)dy = 0 y(0) = 1.
7. Find the general solution to the equation
(x2 − 1)y ′ + 2xy + 3 = 0
8. Show that the relation y = x + ey is an implicit solution to the equation (x + y − 1)y ′ = 1.