ENGINEERING MATHEMATICS – MIDTERM EXAM
Date: April 21, 2021
Name:
ID # (학번):
GRADE (학년):
Rules (Please read the followings very carefully)
➢
Write name and ID #, clearly.
➢
Closed books and notes.
➢
You must show your work in order to receive full credit; correct answers with
no, or insufficient work may not receive credit.
➢
Definitions and formulas for the exam should be used what we had used in
the class.
➢
Your answer must be simplified.
➢
Your answer must be done by both the best effort and the maximum
performance and should always be optimal.
➢
You must write your answer nicely, neatly, and clearly.
➢
No electronic devices including calculators and smartphones are allowed.
➢
No chatting. Please raise your hand if you have any questions.
➢
If anyone is to be considered of committing cheating, then she or he should
be dismissed immediately.
1.
[15-point] Answer either “TRUE” or “FALSE”.
The rules for grading are followings:

+2 and -2 points for the correct and incorrect answer, respectively.

Zero point if no answer
(a) The differential equation
𝑑𝑦
𝑑𝑥
=
𝑥 3 𝑦+𝑥 2 𝑦 2
𝑥 2 +𝑥𝑦
is separable.
(b) A linear differential equation can be rewritten as an exact differential equation.
(c) If 𝑦1 , 𝑦2 , ⋯ , 𝑦𝑛 are solutions to a regular 𝑛𝑡ℎ − 𝑜𝑟𝑑𝑒𝑟 linear homogeneous
differential equation such that its Wronskian, 𝑊 = 0 at some points of 𝐼 and
nonzero at other points of 𝐼, then {𝑦1 , 𝑦2 , ⋯ , 𝑦𝑛 } is a linearly independent set of
functions.
(d) The general solution to the differential equation
2
𝑑𝑦 𝑑2 𝑦
− 4) = 0
(
𝑑𝑥 𝑑𝑥 2
is 𝑦(𝑥) = 𝑐1 + 𝑐2 𝑥 + 𝑐3 𝑒 2𝑥 + 𝑐4 𝑒 −2𝑥 + 𝑐5 𝑥𝑒 2𝑥 + 𝑐6 𝑥𝑒 −2𝑥 .
(e) The general solution to the differential equation
2
𝑑2 𝑦
𝑑𝑦
+ 5) = 0
( 2−4
𝑑𝑥
𝑑𝑥
is 𝑦(𝑥) = 𝑐1 𝑒 2𝑥 cos 𝑥 + 𝑐2 𝑒 2𝑥 sin 𝑥 + 𝑐3 𝑥𝑒 2𝑥 cos 𝑥 + 𝑐4 𝑥𝑒 2𝑥 sin 𝑥 .
(f) An appropriate form of the particular solution for the nonhomogeneous
differential equation 𝑦 ′′ + 𝑦 = 3 sin 2𝑥 is 𝑦𝑝 (𝑥) = 𝐴0 𝑒 𝑗2𝑥 .
(g) An appropriate form of the particular solution for the nonhomogeneous
differential equation 𝑦 ′′ + 9𝑦 = cos 3𝑥 + sin 4𝑥 is 𝑦𝑝 (𝑥) = 𝐴𝑥𝑒 𝑗3𝑥 + 𝐵𝑒 𝑗4𝑥 .
2.
[15] Solve the following differential equations:
𝑑𝑦
(a) [5] <Linear>
3.
𝑥 𝑑𝑥 − 2𝑦 = 2𝑥 2 ln 𝑥 ,
𝑥>0
(b) [5] <Exact>
cosh 𝑥 cos 𝑦 𝑑𝑥 = sinh 𝑥 sin 𝑦 𝑑𝑦
(c) [5] <Exact>
𝑥 − 𝑥𝑦 − 𝑦 ′ = 0
[20] An object of mass 𝑚 falls freely, starting at a point near the earth's surface.
Assuming that the air resistance is proportional to the velocity of the object.
Let 𝑦(𝑡) be the distance travelled by the object at time 𝑡 from the point it was
released, and let the positive 𝑦 direction be downward. Then, 𝑦(0) = 0, and the
velocity of the object is 𝑣(𝑡) =
𝑑𝑦
𝑑𝑡
. Since the object was dropped freely, the initial
velocity is 𝑣(0) = 0. The forces acting on the object are those due to gravity, 𝐹𝑔 = 𝑚𝑔
and the force due to air resistance, 𝐹𝑟 = −𝑘𝑣 where 𝑘 is a positive constant.
(a) [8] Find the differential equation describing the motion of the object by using
Newton’s second law and solve the differential equation.
(b) [2] Calculate the terminal (final) velocity,
𝑣𝑇 = lim 𝑣(𝑡)
𝑡→∞
(c) [3] Sketch the behavior of the velocity as a function of time.
(d) [7] Determine the equation describing the position of the object at time 𝑡.
4.
[15] Determine whether the given set of functions is linearly dependent or linearly
independent on the given interval. The logical and reasonable explanation should be
accompanied with your answer.
(a) 𝑓1 (𝑥) = 𝑒 𝑥 ,
𝑓2 (𝑥) = 𝑒 −𝑥 ,
𝜋
(b) 𝑓1 (𝑥) = cos (𝑥 + 2 ),
𝑓3 (𝑥) = sinh 𝑥 ,
𝑓2 (𝑥) = sin 𝑥 ,
(c) 𝑓1 (𝑥) = 1 + 2𝑥, 𝑓2 (𝑥) = 1 + |2𝑥|,
𝑓4 (𝑥) = cosh 𝑥
(−∞, ∞)
(−∞, ∞)
(−∞, ∞)
(d) 𝑓1 (𝑥) = 𝑘 + 𝑥, 𝑓2 (𝑥) = 𝑘𝑥, 𝑓3 (𝑥) = 𝑘𝑥 2 , (−∞, ∞) where 𝑘 = 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
(e) 𝑓1 (𝑥) = 𝑥𝑒 𝑥+1 ,
𝑓2 (𝑥) = (4𝑥 − 5)𝑒 𝑥 ,
𝑓3 (𝑥) = 𝑥𝑒 𝑥
(−∞, ∞)
5.
[20] Consider a nonhomogeneous differential equation with the initial values
𝑦 ′′ + 3𝑦 ′ + 2.25𝑦 = 𝑔1 (𝑥) + 𝑔2 (𝑥)
(a) [4] Find the homogeneous (general or complementary) solution, 𝑦𝑐 , for the
associated homogeneous differential equation.
(b) [6] When 𝑔1 (𝑥) = −10𝑒 −1.5𝑥 , obtain the corresponding particular solution, 𝑦𝑝1 ,
for the nonhomogeneous differential equation.
(c) [8] Using Cramer’s Rule, find the corresponding particular solution, 𝑦𝑝2 , if
3
𝑔2 (𝑥) = 𝑥 −1 ∙ 𝑒 2𝑥 . [Hint, 𝑒 𝑥 = ∑∞
𝑛=0
1 𝑛
𝑥 ]
𝑛!
(d) [2] Determine the general solution, 𝑦, for the differential equation.
6.
[15] Consider the differential equation
𝑦 ′′ + 𝑦 = 𝑓(𝑥),
y(𝑥0 ) = 𝑦0 𝑎𝑛𝑑 𝑦 ′ (𝑥0 ) = 𝑦1
where 𝑓 is continuous on the interval [𝑎, 𝑏]. If 𝑥0 ∈ (𝑎, 𝑏), find the solution to the
differential equation with the initial values.
ENGINEERING MATHEMATICS – Quiz [1]
Date: March 26, 2019
Name:
ID # (학번):
GRADE (학년):
Rules (Please read the followings very carefully)
➢
Write name and ID #, clearly.
➢
Closed books and notes.
➢
You must show your work.
➢
Definitions and formulas for the exam should be used what we had used in
the class.
➢
Your answer must be simplified.
➢
Your answer must be done by both the best effort and the maximum
performance and should always be optimal.
➢
You must write your answer nicely, neatly, and clearly.
➢
No electronic devices including calculator and cell phone are allowed.
➢
No chatting. Please raise your hand if you have any questions.
➢
If anyone is to be considered of committing cheating, then she or he should
be dismissed immediately.
1.
[15 points] Answer either “TRUE” or “FALSE”.
The rules for grading are followings:

+2 and -2 points for the correct and incorrect answer, respectively.

Zero point if no answer

An extra point for all correct answers
a)
The differential equation
𝑑𝑦
𝑑𝑥
=
𝑥3 𝑦+𝑥2 𝑦2
𝑥2 +𝑥𝑦
is separable.
b) There is a unique integrating factor for a linear differential equation of the form
𝑦 ′ + 𝑝(𝑥)𝑦 = 𝑓(𝑥) .
c)
A linear differential equation can be rewritten as an exact differential equation.
d) The differential equation (𝑦 2 + cos 𝑥 )𝑑𝑥 + 2𝑥𝑦 2 𝑑𝑦 = 0 is exact.
e) The differential equation
f)
The differential equation
𝑑𝑦
𝑑𝑥
𝑑𝑦
𝑑𝑥
− 𝑒 𝑥𝑦 𝑦 = 5𝑥 √𝑦 is a Bernoulli differential equation.
= √𝑥 y + √𝑥𝑦 is a Bernoulli differential equation.
g) A Ricatti equation can be transformed into a Bernoulli equation.
2.
[30] Solve the following differential equations:
a)
[10] <Separable>
(𝑥 − 𝑘)(𝑥 − 𝑚)𝑦 ′ − (𝑦 − 𝑛) = 0, where 𝑘, 𝑚, 𝑛 are
constant with 𝑘 ≠ 𝑚.
3.
𝑑𝑦
b) [10] <Linear>
𝑥 𝑑𝑥 − 2𝑦 = 2𝑥 2 ln 𝑥 ,
c)
𝑦
𝑑𝑥
𝑥
[10] <Exact>
𝑥>0
+ [1 + ln(𝑥𝑦)]𝑑𝑦 = 0, 𝑥 > 0
[40] For the non-exact differential equation
(2𝑦 2 − 9𝑥𝑦)𝑑𝑥 + (3𝑥𝑦 − 6𝑥 2 )𝑑𝑦 = 0
a)
[10] Justify that the given differential equation is not "exact".
b) [15] Find appropriate constants 𝑎 𝑎𝑛𝑑 𝑏 for an integrating factor, μ(𝑥, 𝑦) =
𝑥 𝑎 𝑦𝑏 .
c)
4.
[15] By exploiting the results of b), solve the differential equation.
[15] Solve the Riccati equation,
𝑦 ′ = (𝑦 − 𝑥)2 + 1 ,
where the known solution is 𝑦1 (𝑥) = 𝑥 𝑎𝑛𝑑 𝑦(0) =
1
2
ENGINEERING MATHEMATICS – Midterm Exam
Date: April 22, 2019
Name:
ID # (학번):
GRADE (학년):
Rules (Please read the followings very carefully)
➢
Write name and ID #, clearly.
➢
Closed books and notes.
➢
You must show your work.
➢
Definitions and formulas for the exam should be used what we had used in
the class.
➢
Your answer must be simplified.
➢
Your answer must be done by both the best effort and the maximum
performance and should always be optimal.
➢
You must write your answer nicely, neatly, and clearly.
➢
No electronic devices including calculator and cell phone are allowed.
➢
No chatting. Please raise your hand if you have any questions.
➢
If anyone is to be considered of committing cheating, then she or he should
be dismissed immediately.
1.
[10 points] Determine whether the given set of functions is linearly dependent or
linearly independent on the given interval. The logical and reasonable explanation
should be accompanied with your answer.
a) 𝑓1 (𝑥) = 𝑥 𝑛 ,
𝑓2 (𝑥) = 𝑥 𝑛+1 , 𝑛 = 1, 2, ⋯ , 𝑓𝑜𝑟 (−∞, ∞)
b) 𝑓1 (𝑥) = 𝑒 𝑥 , 𝑓2 (𝑥) = 𝑒 −𝑥 , 𝑓3 (𝑥) = sinh 𝑥 , 𝑓4 (𝑥) = cosh 𝑥 , 𝑓𝑜𝑟 (−∞, ∞)
c)
𝑓1 (𝑥) = cos 2𝑥 , 𝑓2 (𝑥) = sin 2𝑥 , 𝑓𝑜𝑟 (−𝜋, 𝜋)
d) 𝑓1 (𝑥) = ln 𝑥 , 𝑓2 (𝑥) = ln 𝑥 2 , 𝑓𝑜𝑟 (0, ∞)
e) 𝑓1 (𝑥) = 𝑥𝑒 𝑥+1 ,
2.
𝑓2 (𝑥) = (4𝑥 − 5)𝑒 𝑥 ,
𝑓3 (𝑥) = 𝑥𝑒 𝑥 , 𝑓𝑜𝑟 (−∞, ∞)
[15] For homogeneous differential equations:
1
a) [7] Two roots of a cubic characteristic equation with real coefficients are 𝑚1 = − 2
and 𝑚2 = 3 + 𝑗 . What is the corresponding homogeneous linear differential
equation?
b) [8] Find the general solution of 𝑦 ′′′ + 6𝑦 ′′ + 𝑦′ − 34𝑦 = 0 if it is known that 𝑦1 =
𝑒 −4𝑥 cos 𝑥 is one solution.
3.
[15] Consider the differential equation with the initial values,
𝑦 ′′ = 𝑥 + 𝑦 − 𝑦 2 ,
𝑦(0) = −1, 𝑦 ′ (0) = 1
Let us assume that the solution 𝑦(𝑥) exists and is analytic at zero. Then, 𝑦(𝑥) may
possess a Taylor series expansion centered at zero.
Find the first six terms of a Taylor series solution.
4.
[20] Solve the homogeneous differential equation with the boundary values:
𝑦 ′′ + 𝜆𝑦 = 0, 𝑦(0) = 0, 𝑦(𝐿) = 0
a) [10] Find 𝜆 in terms of 𝐿 when 𝜆 > 0.
b) [10] Find the general solution.
5.
[25] Solve the given nonhomogeneous differential equation with the initial values:
𝑦 ′′ − 2𝑦 ′ + 𝑦 = 𝑔1 (𝑥) + 𝑔2 (𝑥) ,
𝑦(1) = 41 𝑎𝑛𝑑 𝑦 ′ (1) = 24
a) [10] When 𝑔1 (𝑥) = 4𝑥 2 − 3, obtain the corresponding particular solution, 𝑦𝑝1 , for
the nonhomogeneous differential equation.
b) [10] Using Cramer’s Rule, find the corresponding particular solution, 𝑦𝑝2 , if
𝑔2 (𝑥) = 𝑥 −1 𝑒 𝑥 .
c)
6.
[5] Determine the general solution, 𝑦, for the differential equation.
[25] Consider the linear second-order differential equation,
𝑦 ′′ + 𝑃(𝑥)𝑦 ′ + 𝑄(𝑥)𝑦 = 𝑓(𝑥)
⋯⋯
(𝐴)
If 𝑦1 (𝑥) and 𝑦2 (𝑥) form a fundamental set of solutions on the interval 𝐼 of the
associated homogeneous equation form of (𝐴), then a particular solution of the
nonhomogeneous differential equation (𝐴) on the interval 𝐼 can be found by the
method of variation of parameter.
a) [15] Prove that if 𝑥 and 𝑥0 are numbers in the interval 𝐼, then the particular
solution is written as
𝑥
𝑦𝑝 (𝑥) = ∫ 𝐺(𝑥, 𝑡) 𝑓(𝑡)𝑑𝑡
𝑥0
where
𝐺(𝑥, 𝑡) =
𝑦1 (𝑡)𝑦2 (𝑥) − 𝑦1 (𝑥)𝑦2 (𝑡)
𝑊(𝑡)
and 𝑊(𝑡) = 𝑊𝑟𝑜𝑛𝑠𝑘𝑖𝑎𝑛 = 𝑊(𝑦1 (𝑥), 𝑦2 (𝑥)) ≠ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 𝑖𝑛 𝐼.
b) [10] Solve the nonhomogeneous differential equation using the results of a):
𝑦 ′′ − 𝑦 = 𝑒 2𝑥 ,
𝑦(0) = 0 𝑎𝑛𝑑 𝑦 ′ (0) = 0
ENGINEERING MATHEMATICS – QUIZ[1]
Date: April 22, 2020
Name:
ID # (학번):
GRADE (학년):
Rules (Please read the followings very carefully)
➢
Write name and ID #, clearly.
➢
You must show your work in order to receive full credit; correct answers with
no, or insufficient work may not receive credit.
➢
Definitions and formulas for the exam should be used what we had used in
the class.
➢
Your answer must be simplified.
➢
Your answer must be done by both the best effort and the maximum
performance and should always be optimal.
➢ You must write your answer nicely, neatly, and clearly.
➢
Please submit your answer sheets to kwangbockyou@ssu.ac.kr
➢
File format: Name_ID#_EM_Quiz1_042220.pdf
➢
You must sign on the following “Honor Statement” and attach to your
answer sheets.
Honor Statement
On my honor as a Soongsil University student, I have neither given nor received
any unauthorized assistance on this exam, nor will I.
Signature:
1. [30] Solve the following differential equations:
a) [10] <Separable>
(𝑥 − 𝑘)(𝑥 − 𝑚)𝑦 ′ − (𝑦 − 𝑛) = 0 , where 𝑘, 𝑚, 𝑛 are
constant with 𝑘 ≠ 𝑚.
𝑑𝑦
b) [10] <Linear>
𝑥 𝑑𝑥 − 2𝑦 = 2𝑥 2 ln 𝑥 ,
c) [10] <Exact>
𝑦
𝑥
𝑥>0
𝑑𝑥 + [1 + ln(𝑥𝑦)]𝑑𝑦 = 0, 𝑥 > 0
2. [25] For the non-exact differential equation
(2𝑦 2 − 9𝑥𝑦)𝑑𝑥 + (3𝑥𝑦 − 6𝑥 2 )𝑑𝑦 = 0
a) [15] Find appropriate constants 𝑎 𝑎𝑛𝑑 𝑏 for an integrating factor,
μ(𝑥, 𝑦) = 𝑥 𝑎 𝑦 𝑏 .
b) [10] By exploiting the results of a), solve the differential equation.
3. [20] If 𝑦1 (𝑥) is a known solution to a Riccati equation, then let the
substitution 𝑢 = 𝑦 − 𝑦1 .
Prove that the substitution 𝑢 = 𝑦 − 𝑦1 can transform the Riccati equation into
a Bernoulli equation.
4. [25] 𝑦1 (𝑥) = 𝑥 is a solution of the 2nd order linear differential equation,
𝑥𝑦 ′′ − 𝑥𝑦 ′ + 𝑦 = 0
Use reduction of order to find a second solution, 𝑦2 (𝑥) in the form of an
infinite series. Do not use the formula.
#1
Quiz[1]- Solutions
#1
Quiz[1]- Solutions
#1
Quiz[1]- Solutions
#2
Quiz[1]- Solutions
Quiz[1]- Solutions
Quiz[1]- Solutions
#3
Quiz[1]- Solutions
#4
ENGINEERING MATHEMATICS – Quiz [2]
Date: April 11, 2019
Name:
ID # (학번):
GRADE (학년):
Rules (Please read the followings very carefully)
➢
Write name and ID #, clearly.
➢
Closed books and notes.
➢
You must show your work.
➢
Definitions and formulas for the exam should be used what we had used in
the class.
➢
Your answer must be simplified.
➢
Your answer must be done by both the best effort and the maximum
performance and should always be optimal.
➢
You must write your answer nicely, neatly, and clearly.
➢
No electronic devices including calculator and cell phone are allowed.
➢
No chatting. Please raise your hand if you have any questions.
➢
If anyone is to be considered of committing cheating, then she or he should
be dismissed immediately.
1. [25-point] Consider the linear differential equation,
𝑑𝑦
𝑥, 0 ≤ 𝑥 < 2
+ 2𝑥𝑦 = 𝑓(𝑥), 𝑦(0) = 0, 𝑤ℎ𝑒𝑟𝑒 𝑓(𝑥) = {
−𝑥,
𝑥≥2
𝑑𝑥
a) [15] Solve the differential equation so that the solution 𝑦(𝑥), is continuous
(1 + 𝑥 2 )
on an (0, ∞).
b) [10] Sketch 𝑦(𝑥)
2. [20] Consider the differential equation
𝑦 ′ = 𝑔(𝑥)ℎ(𝑦)
a) [10] Find the integrating factor μ(𝑥, 𝑦) , which makes the differential
equation be exact.
b) [10] Solve the differential equation,
𝑑𝑦
𝑑𝑥
= −3𝑥 2 𝑦,
𝑦(0) = 1, by using the
above results.
3. [20] Consider a Bernoulli differential equation,
𝑦 ′ + 𝑝(𝑥)𝑦 = 𝑞(𝑥)𝑦 𝑛
Find an appropriate integrating factor, 𝜇 (𝑥, 𝑦), to make the equation be exact.
[Hint: 𝑦 ′ + 𝑝(𝑥)𝑦 = 𝑞(𝑥)𝑦 𝑛 can be rewritten as 𝑦 −𝑛 𝑦 ′ + 𝑝(𝑥)𝑦1−𝑛 = 𝑞(𝑥)].
4. [20] Solve
a) [10] Find all possible values of k so that the given differential equation,
𝑘
𝑑𝑦
(1+𝑦 2 + cos 𝑘𝑦 − 2𝑥𝑦) 𝑑𝑥 = 𝑦(𝑦 + sin 𝑘), is exact.
b) [10] Prove that if 𝑦1 = 𝑒 𝑥 and 𝑦2 = 𝑒 −𝑥
are two solutions of a
homogeneous linear differential equation, then 𝑦3 = cosh 𝑥 and 𝑦4 =
sinh 𝑥 are also solutions of the equation.
5. [20] 𝑦1 (𝑥) = 𝑥 is a solution of the 2nd order linear differential equation,
𝑥𝑦 ′′ − 𝑥𝑦 ′ + 𝑦 = 0
Use reduction of order to find a second solution, 𝑦2 (𝑥) in the form of an
1
𝑛
infinite series. Do not use the formula. [Hint, 𝑒 𝑥 = ∑∞
𝑛=0 𝑛! 𝑥 ].