4.6 #2: Find a particular solution to the second-order differential equation. y + 4y = sec 2t.
|
|
- Loreen Atkins
- 7 years ago
- Views:
Transcription
1 4.6 #: Find a paricular soluion o he second-order differenial equaion y + 4y = sec. Soluion: To find he paricular soluion, we use variaion of parameers. Firs, we need o find a fundamenal se of soluions y and y o he associaed homogeneous equaion y + 4y = 0. The characerisic equaion is given by r + 4 = 0. Thus, r = 4 and so r = ± 4 = ±i. Thus, y () = cos and y () = sin form a fundamenal se of soluions o he homogeneous equaion. Now, we form y p = v y + v y = v cos + v sin. From here here are wo ways o proceed, as menioned in he summary in your book on pages The firs way is o use he formulas given in your book on p. 76 in (6.6). The second way is o go hrough he mehod which gave rise o hese equaions. The second way is more inuiive and doesn require you o memorize formulas; he firs way is probably faser. Le us go hrough boh mehods. Mehod : Using he formulas From he formulas given in (6.6) of your book, we see ha y ()g()d y ()g()d v () =, v () = W () W () where g() can be found by comparing he given differenial equaion wih he equaion in he general form y + p()y + q()y = g() and W () is he Wronskian of y and y. For us, we see ha ( ) ( ) y y W () = cos sin y y = = cos (4) + sin (4) = sin cos and g() = sec. Hence, we ge ha sin sec d v () = = sin cos d To inegrae, we le u = cos, du = sin d. Then v () = / du u = 4 ln u = ln(cos ). 4
2 Also, cos sec d v () = = cos cos d = d =. Therefore, he paricular soluion is given by y p () = v cos + v sin = 4 ln(cos ) cos + sin. We have ha y p = v cos + v sin. So Mehod : No using he formulas y p = v cos + v sin v sin + v cos. Now we se he erms wih derivaives of v and v equal o 0. So, se v cos + v sin = 0. This will form one of he main wo equaions for deermining v and v laer. Now, i follows ha Taking he second derivaive, we ge y p = v sin + v cos. y p = v sin + v cos 4v cos 4v sin. We plug y p ino our differenial equaion o ge sec = y p + 4y p = v sin + v cos 4v cos 4v sin + 4v cos + 4v sin = v sin + v cos. Thus, we ge he sysem of equaions v cos + v sin = 0 v sin + v cos = sec. We muliply he firs equaion by sin and he second equaion by cos o ge he equivalen sysem of equaions v cos sin + v sin = 0 v cos sin + v cos =
3 Adding he wo equaions gives v (sin + cos ) =, and so v =, which implies v =. Hence, v = d =. Now, from he equaion v cos + v sin = 0, we see ha v = v ge ha v = / sin. cos sin cos We inegrae by leing u = cos, du = sin d as in he previous mehod o ge Hence, we ge ha v = ln(cos ). 4 y p () = v cos + v sin = 4 ln(cos ) cos + sin.. So since v = /, we 4.6 #4: Verify ha y () = and y () = ln are soluions o he homogeneous equaion y () + 3y () + y() = 0. Use variaion of parameers o find he general soluion o y () + 3y () + y() =. Firs le us verify ha y () and y () are soluions o he homogeneous equaion. Noe y () =, y () = 3. Hence, we see ha y () + 3y () + y () = ( ) ( ) + = 3 + = 0. So y () is a soluion o he homogeneous equaion. Now we show y () = ln is a soluion. Noe y () = ln +, y () = 3 ln 3 3 = 3 ln 3 3.
4 Hence, y () + 3y + y () = ( 3 ln 3 ) ( ln + ) + ln = ln 3 3 ln ln = 0. So y () is also a soluion o he homogeneous equaion. So he homogeneous soluion is given by y h () = C y () + C y () = C + C ln. Now we use variaion of parameers o find he paricular soluion of y () + 3y () + y() =. We will use he formulas given in your book on p. 76. Thus, we need o calculae he Wronskian of y and y, and we need o know g(). Noe he Wronskian is given by ( ) y y W () = y y ( = ln ) ln + = 3 ln ln = 3. Now, noe g() is he righ-hand side when he equaion is in he form Our equaion is y + p()y + q()y = g(). y () + 3y () + y() = so we need o divide boh sides by. Doing so, we obain he equaion y () + 3 y () + y() = 3. Now, we see ha g() =. Thus, using he formulas given in he book, we ge 3 y ()g() d v () = W () = ln d 3 3 = ln d
5 To inegrae, le u = ln, du = d. Then v () = u du = u = (ln ). Nex, noe Therefore, our paricular soluion is given by Hence, he general soluion is given by y ()g() d v () = W () = d 3 = 3 d = ln. y p () = v ()y () + v ()y () = (ln ) + (ln ) = (ln ). y() = y h () + y p () = C + C ln + (ln ). 4.7 #9: Find a paricular soluion o he differenial equaion x + 4x + 5x = 3 sin, x(0) = 0, x (0) = 3 using undeermined coefficiens. Find and plo he soluion of he iniial value problem. Superimpose he plos of he ransien response and he seady-sae soluion. Use differen line syles or colors o differeniae he curves. Soluion: We ry a paricular soluion of he form We calculae he derivaives of x p : x p () = a cos + b sin. x p() = a sin + b cos, x p() = a cos b sin Plugging x p ino he differenial equaion gives 3 sin = x p() + 4x p() + 5x p () = a cos b sin 4a sin + 4b cos + 5a cos + 5b sin = (4a + 4b) cos + (4b 4a) sin.
6 Equaing coefficiens, we ge he sysem of equaions 4a + 4b = 0 4a + 4b = 3 Adding he wo equaions gives 8b = 3, so b = 3 8. From he firs equaion, we see ha a = b, and so a = 3 8. Thus, x p () = 3 8 cos + 3 sin. 8 Now we find he homogeneous soluion. The associaed homogeneous equaion is x + 4x + 5x = 0. The characerisic equaion is r + 4r + 5 = 0. So r = 4 ± 6 4()(5) = 4 ± 4 Hence, he homogeneous soluion is given by x h () = e ( ) C cos + C sin. Thus, he general soluion o our differenial equaion is = ± i. x() = x p () + x h () = 3 8 cos + 3 ) 8 sin + e ( C cos + C sin. Now we need o find C and C using our iniial condiions. Noe To find C, we ake he derivaive of x(): Thus, x(0) = 0 0 = C C = 3 8. x () = 3 8 sin + 3 ) 8 cos e ( C cos + C sin + e ( ) C sin + C cos. So he soluion is given x (0) = 3 3 = 3 8 C + C C = C = = 8. x() = x p () + x h () = 3 8 cos sin + e ( 3 8 cos 8 sin ).
7 The seady-sae soluion is he par of he soluion ha does no decay when. Noe So he seady-sae soluion is lim x() = 3 8 cos sin x seady-sae () = 3 8 cos + 3 sin. 8 The ransien response is he par of he soluion which quickly decays o 0 as. So he ransien response is given by x ransien () = e ( 3 ) cos 8 8 sin. Now we plo he soluion, he seady-sae soluion, and he ransien response.
1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationChapter 7. Response of First-Order RL and RC Circuits
Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationRC (Resistor-Capacitor) Circuits. AP Physics C
(Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationTransient Analysis of First Order RC and RL circuits
Transien Analysis of Firs Order and iruis The irui shown on Figure 1 wih he swih open is haraerized by a pariular operaing ondiion. Sine he swih is open, no urren flows in he irui (i=0) and v=0. The volage
More informationModule 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationCapacitors and inductors
Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationName: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling
Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: -Solving Exponenial Equaions (The Mehod of Common Bases) -Solving Exponenial Equaions (Using Logarihms)
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationSignal Processing and Linear Systems I
Sanford Universiy Summer 214-215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 14-15, Gibbons
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationRandom Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary
Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationModule 3. R-L & R-C Transients. Version 2 EE IIT, Kharagpur
Module 3 - & -C Transiens esson 0 Sudy of DC ransiens in - and -C circuis Objecives Definiion of inducance and coninuiy condiion for inducors. To undersand he rise or fall of curren in a simple series
More informationVoltage level shifting
rek Applicaion Noe Number 1 r. Maciej A. Noras Absrac A brief descripion of volage shifing circuis. 1 Inroducion In applicaions requiring a unipolar A volage signal, he signal may be delivered from a bi-polar
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationAcceleration Lab Teacher s Guide
Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationBehavior Analysis of a Biscuit Making Plant using Markov Regenerative Modeling
Behavior Analysis of a Biscui Making lan using Markov Regeneraive Modeling arvinder Singh & Aul oyal Deparmen of Mechanical Engineering, Lala Lajpa Rai Insiue of Engineering & Technology, Moga -, India
More informationTHE PRESSURE DERIVATIVE
Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationReturn Calculation of U.S. Treasury Constant Maturity Indices
Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationCommunication Networks II Contents
3 / 1 -- Communicaion Neworks II (Görg) -- www.comnes.uni-bremen.de Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζ-funcion o skech an argumen which would give an acual formula for π( and sugges
More informationPart II Converter Dynamics and Control
Par II onverer Dynamics and onrol 7. A equivalen circui modeling 8. onverer ransfer funcions 9. onroller design 1. Inpu filer design 11. A and D equivalen circui modeling of he disconinuous conducion mode
More informationThe naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1
Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces ime-series smoohing forecasing mehods. Various models are discussed,
More informationEquation for a line. Synthetic Impulse Response 0.5 0.5. 0 5 10 15 20 25 Time (sec) x(t) m
Fundamenals of Signals Overview Definiion Examples Energy and power Signal ransformaions Periodic signals Symmery Exponenial & sinusoidal signals Basis funcions Equaion for a line x() m x() =m( ) You will
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationFull-wave rectification, bulk capacitor calculations Chris Basso January 2009
ull-wave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.
More informationChapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m
Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m
More informationA Curriculum Module for AP Calculus BC Curriculum Module
Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy.
More informationMaking Use of Gate Charge Information in MOSFET and IGBT Data Sheets
Making Use of ae Charge Informaion in MOSFET and IBT Daa Shees Ralph McArhur Senior Applicaions Engineer Advanced Power Technology 405 S.W. Columbia Sree Bend, Oregon 97702 Power MOSFETs and IBTs have
More informationPermutations and Combinations
Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide
More informationAnalogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar
Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 0-7-380-7 Ifeachor
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationImproper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b].
Improper Inegrls Dr. Philippe B. lvl Kennesw Se Universiy Sepember 9, 25 Absrc Noes on improper inegrls. Improper Inegrls. Inroducion In Clculus II, sudens defined he inegrl f (x) over finie inervl [,
More informationLongevity 11 Lyon 7-9 September 2015
Longeviy 11 Lyon 7-9 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univ-lyon1.fr
More informationLecture 2: Telegrapher Equations For Transmission Lines. Power Flow.
Whies, EE 481 Lecure 2 Page 1 of 13 Lecure 2: Telegraher Equaions For Transmission Lines. Power Flow. Microsri is one mehod for making elecrical connecions in a microwae circui. I is consruced wih a ground
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationDoes International Trade Stabilize Exchange Rate Volatility?
Does Inernaional Trade Sabilize Exchange Rae Volailiy? Hui-Kuan Tseng, Kun-Ming Chen, and Chia-Ching Lin * Absrac Since he early 980s, major indusrial counries have been suffering severe muli-laeral rade
More informationNewton s Laws of Motion
Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The
More information11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge
More informationMechanical Fasteners Tensile and Shear Stress Areas
Mechanical Faseners Tensile and Shear Sress reas Lecure 28 Engineering 473 Machine Design Threaded Faseners Bol Threaded fasener designed o pass hrough holes in maing members and o be secured by ighening
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK-2100 Copenhagen Ø, Denmark PFA Pension,
More informationModule 3 Design for Strength. Version 2 ME, IIT Kharagpur
Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:
More informationA Simple Introduction to Dynamic Programming in Macroeconomic Models
Economics Deparmen Economics orking Papers The Universiy of Auckland Year A Simple Inroducion o Dynamic Programming in Macroeconomic Models Ian King Universiy of Auckland, ip.king@auckland.ac.nz This paper
More informationLECTURE 7 Interest Rate Models I: Short Rate Models
LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia email: abreccia@emsbbkacuk
More informationAn Optimal Control Approach to Inventory-Production Systems with Weibull Distributed Deterioration
Journal of Mahemaics and Saisics 5 (3):6-4, 9 ISSN 549-3644 9 Science Publicaions An Opimal Conrol Approach o Invenory-Producion Sysems wih Weibull Disribued Deerioraion Md. Aiul Baen and Anon Abdulbasah
More informationMotion Along a Straight Line
Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his
More informationDebt Policy, Corporate Taxes, and Discount Rates
Commens Welcome Deb Policy, Corporae Taxes, and Discoun Raes Mark Grinbla and Jun iu UCA Firs Version: December 25, 21 Curren Version: November 7, 22 Grinbla and iu are boh from he Anderson School a UCA.
More informationCLASSICAL TIME SERIES DECOMPOSITION
Time Series Lecure Noes, MSc in Operaional Research Lecure CLASSICAL TIME SERIES DECOMPOSITION Inroducion We menioned in lecure ha afer we calculaed he rend, everyhing else ha remained (according o ha
More informationPENSION REFORM IN BELGIUM: A NEW POINTS SYSTEM BETWEEN DB and DC
PENSION REFORM IN BELGIUM: A NEW POINS SYSEM BEWEEN B and C Pierre EVOLER (*) (March 3 s, 05) Absrac More han in oher counries, he Belgian firs pillar of public pension needs urgen and srucural reforms
More informationDistance to default. Credit derivatives provide synthetic protection against bond and loan ( ( )) ( ) Strap? l Cutting edge
Srap? l Cuing edge Disance o defaul Marco Avellaneda and Jingyi Zhu Credi derivaives provide synheic proecion agains bond and loan defauls. A simple example of a credi derivaive is he credi defaul swap,
More informationAP Calculus AB 2007 Scoring Guidelines
AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and
More informationResearch Article Solitary Wave Solutions for a Time-Fraction Generalized Hirota-Satsuma Coupled KdV Equation by a New Analytical Technique
Hindawi Publishing Corporaion Inernaional Journal of Differenial Equaions Volume, Aricle ID 954674, pages doi:.55//954674 Research Aricle Soliary Wave Soluions for a Time-Fracion Generalized Hiroa-Sasuma
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationIntroduction to Option Pricing with Fourier Transform: Option Pricing with Exponential Lévy Models
Inroducion o Opion Pricing wih Fourier ransform: Opion Pricing wih Exponenial Lévy Models Kazuhisa Masuda Deparmen of Economics he Graduae Cener, he Ciy Universiy of New York, 365 Fifh Avenue, New York,
More informationCHAPTER FIVE. Solutions for Section 5.1
CHAPTER FIVE 5. SOLUTIONS 87 Soluions for Secion 5.. (a) The velociy is 3 miles/hour for he firs hours, 4 miles/hour for he ne / hour, and miles/hour for he las 4 hours. The enire rip lass + / + 4 = 6.5
More informationPRESSURE BUILDUP. Figure 1: Schematic of an ideal buildup test
Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics PRESSURE BUILDUP I is difficul o kee he rae consan in a roducing well. This is no an issue in a buildu es since he well is closed.
More informationStochastic Calculus and Option Pricing
Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes
More informationChapter 2 Kinematics in One Dimension
Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how
More informationImagine a Source (S) of sound waves that emits waves having frequency f and therefore
heoreical Noes: he oppler Eec wih ound Imagine a ource () o sound waes ha emis waes haing requency and hereore period as measured in he res rame o he ource (). his means ha any eecor () ha is no moing
More informationTime Consisency in Porfolio Managemen
1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen
More informationDiagnostic Examination
Diagnosic Examinaion TOPIC XV: ENGINEERING ECONOMICS TIME LIMIT: 45 MINUTES 1. Approximaely how many years will i ake o double an invesmen a a 6% effecive annual rae? (A) 10 yr (B) 12 yr (C) 15 yr (D)
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures
More informationNikkei Stock Average Volatility Index Real-time Version Index Guidebook
Nikkei Sock Average Volailiy Index Real-ime Version Index Guidebook Nikkei Inc. Wih he modificaion of he mehodology of he Nikkei Sock Average Volailiy Index as Nikkei Inc. (Nikkei) sars calculaing and
More informationSignal Rectification
9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, half-wae and fullwae. Le s firs consider he ideal
More informationUsefulness of the Forward Curve in Forecasting Oil Prices
Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,
More informationAnalyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective
Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive Alexander Bohner, Nadine Gazer Working Paper Chair for Insurance Economics Friedrich-Alexander-Universiy
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationSimulation of the motion of a sphere through a viscous fluid
ENSEÑANZA REVISTA MEXICANA DE FÍSICA 49 () 166 174 ABRIL 003 Simulaion of he moion of a sphere hrough a viscous fluid R.M. Valladares a, P. Goldsein b, C. Sern c, and A. Calles d Deparameno de Física,
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More informationSHB Gas Oil. Index Rules v1.3 Version as of 1 January 2013
SHB Gas Oil Index Rules v1.3 Version as of 1 January 2013 1. Index Descripions The SHB Gasoil index (he Index ) measures he reurn from changes in he price of fuures conracs, which are rolled on a regular
More informationGoverning the Resource: Scarcity-Induced Institutional Change *
Governing he Resource: carciy-induced Insiuional Change May 8 James Roumasse Nori arui Absrac We provide a dynamic model of naural resource managemen where he opimal insiuional srucure ha governs resource
More informationUNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.
UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL
More informationA Production-Inventory System with Markovian Capacity and Outsourcing Option
OPERATIONS RESEARCH Vol. 53, No. 2, March April 2005, pp. 328 349 issn 0030-364X eissn 1526-5463 05 5302 0328 informs doi 10.1287/opre.1040.0165 2005 INFORMS A Producion-Invenory Sysem wih Markovian Capaciy
More informationFourier Series and Fourier Transform
Fourier Series and Fourier ransform Complex exponenials Complex version of Fourier Series ime Shifing, Magniude, Phase Fourier ransform Copyrigh 2007 by M.H. Perro All righs reserved. 6.082 Spring 2007
More informationChapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More information