MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
Math 267 : Homework 1
Due January 12, 2011
1. Find the general solution of the following DEs. (a) y 2y = 0, (b) y + 5y + y = 0, (c) x2 y = y 2 + xy , [Hint: start with the substitution u = y/x.] 2. Solve the following IVPs and sketch the solutions. y 5y +
Math 267 Final Exam
Apr 23, 2010
Duration: 150 minutes
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Student Number:
Section:
Do not open this test until instructed to do so! This exam should have 17 pages,
including this cover sheet. No textbooks, calculators, or other aids are allowed. One pa
Be sure this exam has 11 pages including the cover
The University of British Columbia
Final Exam December 2009
Mathematics 267, Mathematical Methods for EE and CS Students
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Signature
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This exam consists of 5 questions worth 100 marks in
The University of British Columbia
Final Examination  December 18, 2012
Mathematics 267
Section 101
Closed book examination
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Time: 2.5 hours
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Signature
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Special Instructions:
No books, notes, or calculators are allowed.
Rules
April, 2007
MATH 267
Name
Page 2 of 14 pages
Marks
[15]
1.
(a) An elastic string of length 4 with xed ends has an initial shape u(x, 0) = f (x), where
f (x) =
0
1
0
if 0 x < 1
if 1 x 3
if 3 < x 4
It is released from rest at time t = 0. Assume that the dis
Math 267 Final Exam
Apr 20, 2009
Duration: 150 minutes
Student Number:
Name:
Section:
Do not open this test until instructed to do so! This exam should have 17 pages,
including this cover sheet. No textbooks, calculators, or other aids are allowed. One pa
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
Derivation of the Telegraph Equation
Model an infinitesmal piece of telegraph wire as an electrical circuit which consists of a resistor of resistance R dx and a coil of inductance L dx. If i(x, t) is the current through the wire, the voltage across i the
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
Solution of the Wave Equation by Separation of Variables
The Problem Let u(x, t) denote the vertical displacement of a string from the x axis at position x and time t. The string has length . Its left and right hand ends are held fixed at height zero and
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
Fourier Series
Much of this course concerns the problem of representing a function as a sum of its different frequency components. Expressing a musical tone as a sum of a fundamental tone and various harmonics is such a representation. So is a spectral de
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
Solution of the Heat Equation by Separation of Variables
The Problem Let u(x, t) denote the temperature at position x and time t in a long, thin rod of length that runs from x = 0 to x = . Assume that the sides of the rod are insulated so that heat energy
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
Periodic Extensions
We know that every (sufficiently smooth) periodic function has a Fourier series expansion. It is fairly common for functions arising from certain applications to be defined only on a finite interval 0 < x < . This is the case if, for e
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
Complex Numbers and Exponentials
Definition and Basic Operations
A complex number is nothing more than a point in the xyplane. The sum and product of two complex numbers (x1 , y1 ) and (x2 , y2 ) is defined by (x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y2
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
The Fourier Transform
As we have seen, any (sufficiently smooth) function f (t) that is periodic can be built out of sin's and cos's. We have also seen that complex exponentials may be used in place of sin's and cos's. We shall now use complex exponential
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
Using the Fourier Transform to Solve PDEs
In these notes we are going to solve the wave and telegraph equations on the full real line by Fourier transforming in the spatial variable. We start with The Wave Equation If u(x, t) is the displacement from equi
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
Discretetime Fourier Series and Fourier Transforms
We now start considering discretetime signals. A discretetime signal is a function (real or complex valued) whose argument runs over the integers, rather than over the real line. We shall use square brack
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
DiscreteTime Linear, Time Invariant Systems and zTransforms
Linear, time invariant systems "Continuoustime, linear, time invariant systems" refer to circuits or processors that take one input signal and produce one output signal with the following propert
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
Review of Ordinary Differential Equations
Definition 1 (a) A differential equation is an equation for an unknown function that contains the derivatives of that unknown function. For example y (t) + y(t) = 0 is a differential equation for the unknown funct
MATHEMATICAL METHODS FOR ELECTRICAL AND COMPUTER ENGINEERING
MATH 267

Spring 2010
The RLC Circuit
The RLC circuit is the electrical circuit consisting of a resistor of resistance R, a coil of inductance L, a capacitor of capacitance C and a voltage source arranged in series. We're going to think of the voltage x(t) R + x(t)  L + y(t)
Math 267 Final Exam
Section 101
December 10, 2011
Duration: 150 minutes
Name:
Student Number:
Do not open exam until so instructed.
Pages
Marks
23
14
One 8.5x11 doublesided page of notes is allowed. No textbooks, calculators, or other
aids are allowed.