M351 Sample Hour Examination
Warning: This set of questions is similar to an examination given in a previous year. There is NO GUARANTEE that the questions here will be in any way related to the questions that appear on the actual hour exam for the course
Name (please print):_
Closed book / closed notes.
A bank manager has developed a new system to reduce the time customers spend waiting to be served by tellers during peak business
hours. Typical waiting times h
ANSWERS LINEAR EQUATIONS EXAMPLES
1. dy = y + 5 , dx y (0) = 1 .
The homogeneous problem, y = y , has general solution yh (x) = K ex . Since the forcing term is constant, we look for constant solutions of the nonhomogeneous equation. This equation has the
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EXAMPLES FOR SOLVING INITIAL VALUE PROBLEMS ax + bx + cx = 0 x(t0) = x0, x(t0) = x0.
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EXAMPLES OF HOMOGENEOUS SECOND ORDER LINEAR EQUATIONS
dy d2 y = 2y . 2 dx dx
y 16 y = 0 ,
with the two sets of initial conditions (a) y (1) = 1 , and (b) y (1) = 0 y (1) = 0 . y (1) = 1
z 2 z + z = 0 , with z (0) = 0, z (0) = 1 .
x + 10 x + 24
GUIDELINES FOR THE METHOD OF UNDETERMINED COEFFICIENTS
Given a constant coecient linear dierential equation ay + by + cy = g (x), where g is an exponential, a simple sinusoidal function, a polynomial, or a product of these functions: 1. Find a pair of lin
GUIDELINES FOR THE METHOD OF UNDETERMINED COEFFICIENTS
Given the constant coecient linear dierential equation a x + b x + c x = f (t), where f (t) is an exponential, a simple sinusoidal function, a polynomial, or a product of these functions: 1. Solve the
EXAMPLES OF NON-HOMOGENEOUS SECOND ORDER LINEAR EQUATIONS
d2 y dy 2y = x 2 dx dx
d2 y dy 2y = 3x2 2 dx dx
3. with the initial conditions
y 16 y = sin (2x) , y (1) = 1 , y (1) = 0
given that the general solution of the homogeneous problem is yh (x) =
Solutions for Second Order Examples
(1) d2 y dy = 2y . dx2 dx Characteristic equation: Characteristic roots: General Solution: 2 2 = 0 1 = 1, 2 = 2 y (x) = c1 et + c2 e2t
(2) y 16 y = 0 , with the two sets of initial conditions (a) y (1) = 1 , and (b) y (
Seperable Equations ExamplesANSWERS
dy = 3x2 ey , dx
y (0) = 1 .
Since f (x, y ) = 3x2 ey and ey = 0 for any y , there are no equilibrium solutions. Separating variables, we have ey(x) dy (x) dx = 3x2
and integrating each side with respect to the indep
UNDETERMINED COEFFICIENTS for FIRST ORDER LINEAR EQUATIONS
This method is useful for solving non-homogeneous linear equations written in the form dy + k y = g (x) , dx where k is a non-zero constant and g is 1. a polynomial, 2. an exponential ert , 3. a p
THE CASE OF FIRST ORDER LINEAR CONSTANT COEFFICIENTS: the initial value problem.
THE EQUATION: dx = a x + b(t) dt THE INITIAL CONDITION: x(0) = xo
THE HOMOGENEOUS PROBLEM: dx = ax. dt SOLUTION OF HOMOGENEOUS PROBLEM: xh (t) = K eat .
MATH 351 Exam Hi, WS '14 NAME;
Instructions: Closed book and notes. No cell phones or other aids. Write on one side of paper only.
Show your steps and reasoning to avoid point deductions. For instance, jumping from a differential
equation to r2 3 r + 3 =
MATH 351, Exam 11, WS '14
Instructions: No calculators, cell phones, or aids of any sort. Write only on one side of the paper.
Grading is based both on the nal "answer," and on how clearly you show your steps and reasoning.
Arrange problems in n
MATH 351, Exam IV, WS13
INSTRUCTIONS: Closed book, closed notes, no cell phones or calculators or any
other aids. Show your steps and reasoning; write clearly, neatly, and dark enough, using
proper mathematical notation.
1. Derive the general solution of
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Closed book, closed notes, no cell phones or ay other aids used. Use only this paper (I
can give more if needed), Must show your steps and reasoning and use prOper
i 1. Solvethe differential equation ,y_
MATH 351, Exam I, WS 2013
Closed book, closed notes, no calculators or aids of any kind. Use only this paper. (I can
give you more if you need it.) Show your steps and reasoning, be sure to use proper math
notation (for example, dont neglect to inclu
M351 Sample Hour ExaminationANSWERS
1. (a) For 1 dx x = t2 , t > 0 , dt t the associated homogeneous equation is dx dt 1 t x = 0.
(b) The general solution of the homogeneous equation is
= C exp = Ct.
1 ds s
= C eln (t)
32 t ,
Department of Mathematical Sciences University of Delaware Prof. T. Angell September 23, 2009
Exercise Sheet 3 Exercise 11: A growth model that is used in such diverse elds as Economics and Medicine (in this latter case, to model the size
Department of Mathematical Sciences University of Delaware Prof. T. Angell October 9, 2009
Exercise Sheet 4
Exercise 16: A particle moves in a straight line in such a way that its distance x from the origin at time t obeys the dierential e
Department of Mathematical Sciences University of Delaware Prof. T. Angell October 21, 2009
Exercise Sheet 5 Exercise 21: For the forced damped oscillator x + 2 x + 26 x = 82 cos (4 t) (a) Find the transient solution. (b) Find the steady s
Department of Mathematical Sciences University of Delaware Prof. T. Angell November 4, 2009
Exercise Sheet 6 Exercise 26: (a) Solve the two-point boundary value problem y + 2 y + y = 0 , y (0) = 2 , y (2) = 2 .
(b) Consider the two-point b
Department of Mathematical Sciences University of Delaware Prof. T. Angell November 11, 2009
Exercise Sheet 7 Exercise 31: Compute the inverse of the following using elementary row operations 12 1 cos () 0 sin () . 3. 0 1 0 (a) A = 2 1 (b)