MATH 2280-002
Exam 1 Extra Credit (#7)
Math 2280 Section 002 [SPRING 2013]
Instructions: This is extra credit. It’s optional and won’t hurt your grade whether or not you
attempt it. The points you score on this assignment will be used to augment your score on question
#7 of Exam 1. Regardless of how well you perform on this assignment, you will be allowed to score
at most 7 out of 10 points on #7. In other words, if you scored 4/10 on #7, you can earn at most
3 points on this assignment. I have indicated how much each problem is worth so that, if you do
not need many points, you will not need to do as much work. You can attempt as many problems
as you like.
This assignment is due in class on Friday, March 8. You will receive either full or no credit on each
problem. I will be grading more harshly than usual because this is extra credit and I’m anticipating
a lot of questions to grade. If you don’t show enough work, you will receive no points. You need to
convince me that you didn’t just plug the differential equation into Maple or copy another student’s
work. If your work is too messy for me to follow easily, you will receive no points. Be sure to clearly
indicate your final answer. Don’t included unrelated scratch work.
For problems 1-6, find the general solution to the given DE using the integrating factor method. Be
sure to clearly mark your integrating factor.
1. ( 13 pt) sec(x)y 0 = y − 1
2. ( 13 pt) 10y 0 = 5y + cos(x)
y
3. ( 13 pt) y 0 − 8x +
=0
(10 + 3x)
4. ( 13 pt) y 0 + 2xy − 4x = 0
p
y
5. ( 31 pt) y 0 + = 3 x2 − 9
x
6. ( 13 pt) y 0 + tan(x)y = tan(x)
For problems 7-11, find the particular solution to each DE using the integrating factor method and
the initial condition given. Be sure to clearly mark your integrating factor.
7.
8.
9.
10.
11.
( 13
( 31
( 13
( 31
( 13
pt)
pt)
pt)
pt)
pt)
y 0 − 2y = 3e2x , y(0) = 2
2xy 0 + 3y = 2x3 , y(9) = −1
(x2 + 5)y 0 + 3xy = x, y(2) = 1
xy 0 + (2x − 3)y = 3x4 , y(1) = 1
(x2 + 1)y 0 + 3x3 y = 6x exp(− 32 x2 ), y(0) = 2
12. ( 23 pt) A tank initially contains 100 gal of pure water and 50 lb of salt. Brine containing 2 lb
of salt per gallon enters the tank at the rate of 4 gal/s, and the well-mixed brine in the tank
flows out at the rate of 6 gal/s.
(a) Give an equation for the volume of the solution in the tank at time t. Simplify as much as
possible.
(b) Give an equation for the salt concentration of the solution at time t in terms of the amount
of salt x(t). Simplify as much as possible.
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MATH 2280-002
Exam 1 Extra Credit (#7)
(c) Give an equation for the amount of salt in the tank at time t. Simplify as much as possible.
13. ( 31 pt) A tank initially contains 400 gal of pure water and 100 lb of salt. Brine containing 2
lb of salt per gallon enters the tank at the rate of 4 gal/s, and the well-mixed brine in the tank
flows out at the rate of 4 gal/s.
(a) Give an equation for the volume of the solution in the tank at time t. Simplify as much as
possible.
(b) Give an equation for the salt concentration of the solution at time t in terms of the amount
of salt x(t). Simplify as much as possible.
(c) Give an equation for the amount of salt in the tank at time t. Simplify as much as possible.
14. ( 31 pt) A tank initially contains 300 gal of pure water and 100 lb of salt. Brine containing 3
lb of salt per gallon enters the tank at the rate of 4 gal/s, and the well-mixed brine in the tank
flows out at the rate of 7 gal/s.
(a) Give an equation for the volume of the solution in the tank at time t. Simplify as much as
possible.
(b) Give an equation for the salt concentration of the solution at time t in terms of the amount
of salt x(t). Simplify as much as possible.
(c) Give an equation for the amount of salt in the tank at time t. Simplify as much as possible.
15. ( 31 pt) A tank initially contains 150 gal of pure water and 100 lb of salt. Brine containing 4
lb of salt per gallon enters the tank at the rate of 6 gal/s, and the well-mixed brine in the tank
flows out at the rate of 5 gal/s.
(a) Give an equation for the volume of the solution in the tank at time t. Simplify as much as
possible.
(b) Give an equation for the salt concentration of the solution at time t in terms of the amount
of salt x(t). Simplify as much as possible.
(c) Give an equation for the amount of salt in the tank at time t. Simplify as much as possible.
16. ( 32 pt) Suppose there are two cascading tanks (as in Figure 1.5.5 of your textbook). Let Tank
1 and and Tank 2 have volumes 150 gal and 300 gal, respectively. Suppose each tank initially
contains 100 lb of salt. The flow rate into Tank 1, the flow rate out of Tank 1 (and into Tank
2), and the flow rate out of Tank 2 are each 10 gal/min. Pure water is flowing into Tank 1.
(a) Find the amount x(t) of salt in Tank 1 at time t.
(b) Find the amount y(t) of salt in Tank 2 at time t.
(c) Find the maximum amount of salt ever in Tank 2.
17. ( 23 pt) Suppose there are two cascading tanks (as in Figure 1.5.5 of your textbook). Let Tank
1 and and Tank 2 have volumes 100 gal and 400 gal, respectively. Suppose each tank initially
contains 100 lb of salt. The flow rate into Tank 1, the flow rate out of Tank 1 (and into Tank
2), and the flow rate out of Tank 2 are 5, 6, and 10 gal/min, respectively. Pure water is flowing
into Tank 1.
(a) Find the amount x(t) of salt in Tank 1 at time t.
(b) Find the amount y(t) of salt in Tank 2 at time t.
(c) Find the maximum amount of salt ever in Tank 2.
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MATH 2280-002
Exam 1 Extra Credit (#7)
18. ( 32 pt) You obtain a job with a starting salary of $30, 000 per year. Your salary S(t) increases
exponentially, with S(t) = 30et/20 thousand dollars after t years. Meanwhile, 10% of her salary
is deposited continuous in a retirement account, whcih accumulates interest at a continuous
annual rate of 6%.
(a) Derive a differential equation for the amount A(t) in her retirement account after t years.
(b) Compute the amount in her retirement account after 30 years.
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