Math 2214 Test 1 Sample Problems
1. Solve the following initial value problems:
(a) e4ty0+ 4e4ty= 1, y(0) = 2
(b) y0=−y+ety2,y(−1) = −1
(c) y0+1
ty=e−t2,y(1) = 12
(d) y0+1
ty=t·y−1,y(2) = 3
2. Solve the initial value problem
ty0+ (t+ 1)y=t, y(ln 2) = 1
and find lim
t→∞ y(t)
3. Consider two initial value problems below:
(t3+ 1)y·y0+t2= 0, y(0) = 1 and y0= 2tp1−y2, y(0) = 0
For each problem do the following:
(a) Find all equilibrium solutions
(b) Find the biggest open rectangle (a<t<band α < y < β) on which the solution of the initial
value problem is guaranteed to exist and be unique. (Hint: For a differential equation y0=f(t, y)
you need to find a region where f(t, y) and ∂f
∂y (t, y ) are continuos as functions of tand y)
(c) Solve the initial value problem. You may leave the solution in the implicit form.
4. The rate of change of the temperature θof an object is described by differential equation θ0=k(S−θ),
where Sis the temperature of the environment.
(a) A pizza is taken from the oven where the temperature was set to 400◦F and placed on the table
in the room where a temperature of 72◦F is maintained. After 10 minutes, the temperature of the
pizza decreased to 236◦F. You cannot eat pizza that is hotter than 154◦F. How much time will it
take until the pizza is safe for eating?
(b) Suppose you are about to be late for a very important meeting and you cannot wait this long. You
decided to put the pizza into a freezer right out of the oven. The temperature inside the freezer
is 0◦F. Using the value of kyou have obtained in the previous problem, write the new differential
equation for this situation and determine the time you will need to wait for the pizza to cool down
to 154◦F.
