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FULL Name
Social Security Number FINAL EXAMINATION
22B, 3:104:00 pm, Friday March 22, 2002
Declaration of honesty: I, the undersigned, do hereby swear to uphold the highest standards of academic honesty, including, but not limited to, submitting work that is original,
my own and unaided by notes, books, calculators, mobile phones, bread machines, mp3
players, handheld toaster ovens, razor scooters, London buses, personal cooling systems
or any other electronic device. Signature Date 1 2 3 4 5 6 TOTAL 1 Advice This examination is divided into two parts: Part A asks you to employ the methods
studied during the course to solve various initial value problems. Do it ﬁrst and
carefully – emphasis will be placed on correct answers. Part B involves harder
questions where you can display your understanding of the material. Partial credit
will only be assigned for steps that genuinely contribute to the solution and will be
limited. Therefore, you are advised to use your time doing a few questions well,
rather than aiming for partial credit for all questions. 2 A Skills Questions
Question 1 (100 points) Solve the following initial value problems using your
favorite method. In each case, indicate whether your solution is the unique one.
If it is not, explain why and ﬁnd all solutions. The majority of the credit will be
awarded for the correct answer with some work. If practical, check that each of
your solutions satisﬁes the original differential equation and initial conditions. ©
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Question 2 (100 points) Raindrops keep falling on my head. Worse still, its very
misty so they get bigger as they fall (the scientiﬁc term is accretion). In this
problem we will show (under simplifying assumptions) that the acceleration of an
.
accreting raindrop is © § ¥ £ ¡
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0 1. Introduce some variables: call
the raindrop’s mass at time .
Assume that it remains spherical at all times so call its radius
and
use the variable
for its velocity. Draw a picture of the raindrop
showing all the variables and the forces acting on it. $ ¥ 0
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. Isaac
raindrop is the product of its mass and velocity,
Newton says that the rate of change of momentum is equal to the sum of
the forces. Assume that air resistance is negligible so the only force is
that of gravity
. Translate this statement into a differential
equation involving
,
and . !%$ ¨$§
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drop is constant. Write down the relation between the mass and radius of
the spherical raindrop. Call the constant of proportionality
.
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9 4. Lets assume that the rate of change of the mass of the raindrop is proportional to its cross sectional area multiplied by its velocity. Translate this
statement into a differential equation. Call this constant of proportionality
.
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# 9 5. Combine the equations you found in parts 3 and 4 above to produce a simple
equation relating the rate of change of the radius to the velocity. 6. Now we want to write the equation of motion found in part 2 in terms of
and
only. Do this using: (i) The massradius relation
the variables
found in part 3 to eliminate
. (ii) Your result for the rate of change
of the mass in part 4. (iii) The chain rule along with the result for the rate
of change of the radius found in part 5 to eliminate any remaining time
derivatives. 0
A 5 A A 7. If all went well you now have a differential equation involving the variables
and (and a bunch of constants) in which is now regarded as an unknown function of the independent variable . Show that this equation is
linear in the variable . 8. Solve the differential equation for
found in part 7. As initial condition,
assume that the velocity is zero when the radius of the raindrop vanishes
(raindrop formation). 9. The last step. Return to the equation of motion in part 2 because it tells you
the rate of change of the velocity (or in other words ACCELERATION).
You already know how to eliminate the mass and its rate of change (using
parts 3 and 4), but now you can also substitute your solution for the velocity
in terms of the radius from part 8. Then you can solve for the acceleration
and, bloopers notwithstanding, are now in seventh heaven! Question 3 (40 points) Psychologists studying mathematics students noticed the
following strange phenomena:
1. If the students knew no calculus and then were locked in a padded cell with
a copy of Barcellos and Stein, their knowledge of calculus remained zero
for all later times.
6 2. If the students knew a little bit of calculus, (i.e. less than that in Barcellos
and Stein), then after sufﬁciently long in the padded cell, their knowledge
of calculus was very close to that in the textbook.
3. Students whose calculus knowledge coincided with the contents of Barcellos and Stein, maintained that level for all later times.
4. Students who knew more calculus than the contents of Barcellos and Stein
eventually reverted back to a knowledge level close to that of the textbook.
Develop a mathematical model describing the psychologist’s results based on a
ﬁrst order autonomous ordinary differential equation. (Your answer should include two graphs, the ﬁrst representing the information in points 14 above, the
second sketching any functions of the unknown variable that appear in your proposed differential equation.) Question 3 working Question 4 (80 points) 9 plane for the system of equations in part
a) Sketch the direction ﬁeld in the
4 of question 1. Draw several trajectories for various initial conditions including
the one you obtained explicitly in question 1. b) Diagonalize the matrix 7 C
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are solutions to 1
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b) Suppose the Wronskian
. Prove that you can always ﬁnd constants
and
such that the
solution
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1 c) Rewrite the second order differential equation in part a) as a pair of ﬁrst order
equations. This system has two solutions. When are they independent? Show that
your answer is equivalent to the one for part a). Question 6 (80 points) §
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1 d) Solve the initial value problem using the Laplace transform method. Question 6 working 10 § 1 c) What is ...
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This note was uploaded on 05/19/2009 for the course MATH 22B taught by Professor Hunter during the Spring '08 term at UC Davis.
 Spring '08
 Hunter

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