1) a) Determine the of the following differential equations. DO NOT SOLVE.
Differential Equation
TYPE
xyy' = 0.
Be^anlLi., r = - *
n/- ftemoulli ,/ n ~ -
iv) y(2x - y) + y' = 1 + xj
v)
b) Solve equation ii),
4f-f^
\_
a
I =t
1
cx
i^
D
"T
X*i
y
D- C
A
*
^ S

/
Qs
MATH 242, QUIZ23, isms/2013 NAME:
Time: 20 minutes LAST NAME:
Question: (i) Determine a suitable integrating factor 0(a) or 0(y) for
d3: + (ta1"r as) dy = 0 (1)
such that the equation becomes exact by multiplying every term with it.
(ii) Find the gen

MATH 242, QUIZALA, 03/06/2013 NAME:
Time: 20 minutes LAST NAME:
Question: Obtain the general solution (general homogeneous plus a particular solution)
using the method of variation of parameter for
may + $211, 9mg = 2.
An er: . r
SW TIM; i) a (quahb Eula

MATH 242, QUIZLB, 13/06/2013 NAME:
Time: 20 minutes LAST NAME:
Question: (i) Determine the general solution of
2
:_= 2
31194; 53:.
(i) Find the solution that satises the initial condition y(l) : 2 and give its region of
validity.
Answer:
. _ A
0) ya); eaf

'2. )
MATH 242, QUIZ-2.A, y5/06/2013 NAME:
Time: 20 minutes LAST NAME:
Question: (i) Determine a suitable integrating fagtor 0(m) or 0(y) for
dw+($ey)dy=0 (1)
such that the equation becomes exact by multiplying every term with it.
(ii) Find the general so

MATH 242, QUIZ-LA, 13/06/2013 NAME:
Time: 20 minutes LAST NAME:
Question: (i) Determine the general solution of
3
I _ :62.
y+33y 55'
(i) Find the solution that satises the initial condition y(1) = 1 and give its region of
validity.
Answer:
X51 (Jw'w) (6",

MATH 242, MT Exam NAME:
Time: 2 Hours, 05.07.2013 LAST NAME:
Q1: (1) (15 pts) Determine the general solution yr) of the linear, non-homogeneous
equation with varying coefcients
$234 my, + 23; = 51:.
(ii) (10 pts) Write down a differential equation for whi

MATH225
Dierential Equations and Linear Algebra (02)
15 minutes (show all your work!)
Quiz #1
September 29, 2015
Name & Last Name:
Department:
Question (10 points) Verify that if c is a constant then the function dened piecewise by
if x c ,
+1
y(x) = cos(

MATH 242, QUIZ-S-A, 12/07/2013 NAME:
Time: 20 minutes LAST NAME:
Question: (1) Find a linear approximation of the following function valid in the indi
cated intervals:
f($ay)=ym3-$y+m2y~1, 1S$31,03y$2.
I 22422 227
(ii) Give a bound on the error of approx

MATH 242, QUIZ43, 03/06/2013 NAME:
Time: 20 minutes LAST NAME:
Question: Obtain the general solution (general homogeneous plus a particular solution)
using the method of variation of parameter for
may" [ mzy' -~ 4339! = 1.
Answer: TLLD S a; Courting? Evtu

gel/mill 6319) M)
MATH 242, QUIZ-3A, 28/06/201? NAME:
Time: 20 minutes LAST NAME:
Question: (i) Determine the general solution of
332g my' + y = 0.
(ii) Write down the general solution of the fthorder, constantcooicient, linear, horno~
geneous different

BiLKENT UNIVERSITY
Department of Mathematics
Date: 22 October 2009 NAME: . .
Time: 18:00-20:00 STUDENT NO: . .
Fall 2009-10, Y. Kurtulmaz 8!, U. Mugan SECTION: 01 02 03 04
Math 225.01-04, Linear Algebra 85 Differential Eq. Midterm Exam # 1
1 2 3 4 5 TOTAL

Economics 11: Solutions to Homework 2
1. Utility Functions (4 points)
Define the utility functions and draw the indifference curves for each of the following cases:
a) Every time I consume one unit of x1 , I want to consume 2 units of x2 .
b) If the price

Economics 11: Solutions to Second Midterm
Instructions: The test is closed book. Calculators are allowed.
Short Questions (25 points)
Question 1
The substitution effect is negative. Assuming there are two goods, this means that the Hicksian
demand h1 (p1

Economics 11: Solutions to Homework 8
November 19, 2009
1. Cost Minimisation Problem (5 points)
A firm has production function
1/2
f (z1 , z2 ) = z1 (z2 1)1/2
The prices of the inputs are r1 and r2 .
(a) Find M P1 , M P2 , and M RT S.
(b) If z2 is fixed a

Economics 11: Solutions to Homework 5
1. Expenditure Minimisation (5 points)
Suppose that an individual has utility u(x1 , x2 ) = x1 (1 + x2 ). Throughout assume income m is
sufficiently high so there is an internal solution.
a) Find the Marshallian deman

Economics 11: Solutions to Homework 7
November 12, 2009
1. Production with One Input (4 points)
Digging clams by hand in Sunset Bay requires only labour input. The total number of clams
obtained per hour (q) is given by: q = 100L1/2 Where L is he labour i

MATH 242, QUIZ-5-B, 12/07/2013 NAME:
Time: 20 minutes LAST NAME:
Question: (i) Find a linear approximation of the following function valid in the indi
cated intervals:
(ii) Give abound on the error of approximation. cfw_1 (12 m? M; , i
f eeeeeee -"%-q=~-w

SOKM/ @J/1 /L&/;
MATH 242, QUIZ-3. B, 28/06/2013 ME:
Time: 20 minutes LAST NAME:
Question: (i) Write down the general solution of the fth-order, constantcoefcient,
linear, homogeneous differential equation if the roots of its characteristic equation are

BILKEN T UNIVERSITY
Mathematics Department
Math225 Differential Equations & Linear Algebra
Fall Semester 2015-2016
FIRST HOMEWORK ASSIGNMENT
October 13, 2015
Due Date: October 20, 2015
IMPORTANT
o This homework consists of 5 questions of equal weight. Hom

BiLKENT UNIVERSITY
Mathematics Department
Math225 Di'erential Equations & Linear Algebra
Summer School 20142015
SECOND HOMEWORK ASSIGNMENT
June 22, 2015
Due Date: June 25, 2015
Name : .
Id. No. : .
Section : .
IMPORTANT
This homework consists of 5 q

Date: 23 July 2004, Friday
Time: 9:30-11:30
Alexander Goncharov & Ali Sinan Sertoz
Math 102 Calculus II Final Exam Solutions
Q-1) Test the following series for convergence:
X
2n
i)
n15
n=1
ii)
X
n=2
n
(1 +
n2 )(ln n)2
15
an+1
n
=2
Solution: i)
2 as n

Math 101
Midterm II
December 8, 2007
10:00 - 12:00
Name
:
ID#
:
Department
:
Section
:
Instructor
:
The exam consists of 5 questions of equal weight.
Please read the questions carefully.
Show all your work in legibly written, well-organized mathematica

Math 101 Spring 2009
Midterm 2 Solutions
1) Find the limit
1
lim
x0 x4
Z
x
tan3 (u) du
0
Solution: We have
1
lim 4
x0 x
Z
Rx
x
tan3 (u) du = lim
x0
0
0
tan3 (u) du
= ()
x4
We have a indeterminate form of type 00 . Lets try applying LHopitals Rule then
by

Date: 21 June 2003, Saturday
Instructor: Ali Sinan Sertoz
Math 102 Calculus Midterm Exam I
Solutions
Z
Q-1) Evaluate the integral
x2 + 5
dx
(x 1)2 (x2 + 1)
Solution: Here you need to simplify the integrand using the technique of partial fractions:
Cx + D

Name Surname:
Signature:
October 4, 2012
MATH 101-006 Quiz 2
a) (5 points) Show that the equation cos x = x has at least one root in the open interval
(0, /2). Explain your solution and write the name of the theorem that you use.
b) (5 points) The followi

Name Surname:
Signature:
October 11, 2012
MATH 101-005 Quiz 3
a) Suppose you want to send an e-mail to me or to the coordinator of MATH 101. Write
down three things that your e-mail must contain.
1.
2.
3.
b) Write the equation of the tangent line to the g