Homework #3 Math 527

Homework #3 Math 527

Follow the usual instructions on homework submission: Be clear, legible, and organized.
Write on loose-leaf paper. Staple together in the upper-left-hand corner, write your name,
section #, Math 527 HW2, and date in the upper-right-hand corner.
Problems 1-3. Find the general solution. The “prime” notation indicates differentiation
with respect to x: y
0 = dy/dx, etc.
1. 6y
00 − 7y
0 + y = 0
2. y00 + 2y
0 + 3y = 0
3. y00 − 6y
0 + 9y = 0
Problems 4-6. Solve the initial-value problem. As in problems 1-3, y
0 = dy/dx, etc.
4. 2y
00 + y
0 − 10y = 0, y(1) = 5, y0
(1) = 2
5. 4y
00 − 4y
0 + y = 0, y(0) = 0, y0
(0) = 3
6. y00 + y
0 + 2y = 0, y(0) = 1, y0
(0) = −2
Applications of 1st-order equations
Problem 7. A thermometer is removed from a room where the temperature is 70◦
F and is placed outside where the air temperature is 10◦ F. Thirty seconds later, the
thermometer reads 50◦ F. What is the reading on the thermometer one minute after
being taken outside? How long will it take for the thermometer to reach 15◦ F?
Approach: use Newton’s law of cooling
dT
dt = k(T − Tm)
where T(t) is the temperature of the thermometer, Tm is the ambient temperature (here
Tm = 10), and k is a constant of proportionality. Solve the differential equation for T(t)
and then use the initial condition T(0) = 70 and thirty-second reading T(30) = 50 to
determine the values of the constants.
Once you have the explicit solution T(t), go back and answer the questions in the problem
statement.
Problem 8. A tank contains 200 liters of water in which 30 grams of salt is dissolved.
Brine containing 1 gram of salt per liter is pumped into the tank at a rate of 4 L/min.
The well-mixed solution is pumped out at the same rate. Derive a differential equation
for A(t), the number of grams of salt in the tank at time t. Then solve the differential
equation to get an explicit function for A(t). How many grams of salt will be in the tank
as t → ∞?
Approach: dA/dt = Rin − Rout, where Rin is the rate of salt flowing in, and Rout is the
rate of salt flowing out, both in grams per min. A little thinking yields
Rin = 1 g/L · 4 L/min = 4 g/min
and
Rout =
A(t) g
200 L
· 4 L/min = A
50
g/min
Drop the g/min units to get Rin = 4 and Rout = A(t)/50. Plug these back into the
equation for dA/dt. Then solve the differential equation and initial value problem.
Once you have the explicit solution A(t), go back and answer the questions in the problem
statement.
Problem 9. Two chemicals A and B combine in reaction to form a chemical C. The
rate of production of C is proportional to the amounts of chemicals A and B present at
any given instant. Two grams of A combine with one gram of B to form three grams
of C. Initially, there are 100 grams of A, 50 grams of B, and 0 grams of C. Ten minutes
later, 10 grams of C have been formed. Derive a function C(t) for the number of grams
of C at time t. How many grams of C are formed in the limit of infinite time? How long
does it take for half that amount of C to be produced?
The reaction is governed by the relation
dC
dt = kA(t)B(t)
One gram of C is composed of 2/3 grams A and 1/3 gram B. Given the initial concentrations of A,B,C, a little thinking produces these equations: A(t) = 100 − 2/3 C(t) and
B(t) = 50 − 1/3 C(t). Plug those into the equation for dC/dt and solve the resulting
initial value problem.
Once you have the explicit solution C(t), go back and answer the questions in the problem
statement.
Problems 7 and 8 are taken from Zill’s textbook, exercises 3.1 #13 and #20.