University at Albany Differential Equation Midterm Exam Questions 13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE question

University at Albany Differential Equation Midterm Exam Questions 13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions13 DE questions International Association of University Summer Sessions
Midterm Exam
Differential Equation
July 15th, 2020
Mr. Hernández
Student’s Name:
Student Instructions:
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This exam has three separate parts for a total of 100 points:
a) True & False and Fill-in the blank section with 4 questions worth 7.5 points each.
b) Straight forward exercises with 4 questions worth 7.5 points each.
c) Open-ended section with 4 questions worth 10 points each
Questions will always receive credit based on accuracy, justification of work, and
completeness of the answer.
The formulas and identities that you may need to answer some questions are found at the
end of the examination on a reference sheet.
Show your work, even when you are using a calculator.
Part I: True and False question. Determinate whether the statement is true or false. If it is False, justify
your answer.
1)_______ The DE
??2 ??
????2
???? 2
????
= ?1 + ( ) is a linear equation.
2) _______ Every separable first-order equation
????
?????
???? = ??(??)?(??) is exact.
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3)_______ The initial-value problem ???? = ?? , ??(0) = 0 doesn’t have solution.
4) _______ A constant multiple of a solution of a linear differential equation is also a solution.
PART II: Exercises. Please show your work even if you use calculator.
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5) Given that ?? = ?? 2 +?? is a one-parameter family of solutions of the first-order DE ?? ? + 2???? 2 = 0. Find a
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solution of the first-order initial value problem consisting of this DE and the initial value ??(?2) = 2. Give the
largest interval I over which the solution is defined.
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????
6) Given (?? + 1) ???? = 2???? ??? ? (?? + 2)??, find the general solution and give the largest interval ?? over which
the general solution is defined.
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7) Solve cos ?? ???? + (1 + ??) sin ?? ???? = 0
8) Solve ?? ?? ? 4?? ? + 4?? = (?? + 1)?? 2?? .
Part III. Open ended questions. All work must be shown.
9) A cup of coffee cools according to Newton’s law of cooling. Use data from the graph of temperature T(t) in the figure
????
below to estimate the constants ???? , ??0 , and ?? in a model of the form of a first-order initial-value problem: ???? =
??(?? ? ???? ), ??(0) = ??0 .
10) A portion of a uniform chain of length 8 ft is loosely coiled around a peg at the edge of a horizontal
platform as you can see in the figure, and the remaining portion of the chain hangs at rest over the edge of
the platform. When all resistive forces are ignored, it can be shown that a mathematical model relating the
velocity ?? that the chain fall to the floor to the length ??(??) of the chain over-hanging the table at time t>0,
????
is given by ???? ???? + ?? 2 = 32??.
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Rewrite the DE and then solve it using the fact that it’s a Bernoulli equation.
11) A series circuit is shown in the figure. The values of Capacitance and resistance are ½ henry, 10 ohms respectably.
Determinate the current ?? if the initial current is zero.
12) Use Laplace transform to solve each system.
?? ?? + ?? ?? = ?? 2??
2?? ? + ?? ?? = ??? 2??
Where ??(0) = ??(0) = ?? ? (0) = ?? ? (0) = 0.
Extra points
The differential equation (?? ? ??? 2 + ?? 2 )???? + ?????? = 0 is not exact, show how the rearrangement
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(?????? + ??????)/ ??? 2 + ?? 2 = ???? and the observation ??(?? 2 + ?? 2 ) = ?????? + ?????? can lead to a solution.
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