Math 201 Lecture 12: Cauchy-Euler Equations
|
|
- Shanon McGee
- 7 years ago
- Views:
Transcription
1 Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem number in he 8h ediion. To solve general 2nd order linear equaions, 0. Review a() y +b() y + c() y = f(). (). Guess one soluion y. (Popular guesses: consans; exponenials; simple polynomials; sin,cos) 2. Wrie he equaion ino sandard form y + p() y + q() y = 0 (2) and apply he reducion of order formula: e p()d y 2 ()= y () y () 2 d (). Apply variaion of parameers formula wih Quiz: Solve v = y p = v y + v 2 y 2 (4) f() y 2 () a()[y y 2 y y 2 ], v 2 = o obain y p. Noe ha a() is no a consan anymore. 4. The general soluion is hen given by Simplify if possible. 5. Check your soluion if ime allows. (As = 0 is a singular poin, le s jus consider > 0). Soluion.. Guess one soluion for I is clear ha any consan is a soluion. So le s ake y =. 2. Use reducion of order o ge y 2. a. Wrie he equaion in sandard form: f() y () a()[y y 2 y y 2 ]. (5) y =C y + C 2 y 2 + y p. (6) y y =. (7) y y =0. (8) b. Compue y y = p()=. (9) e p y 2 = y 2 = e = e ln = = 2 y 2. (0). Use variaion of parameers o ge y p.
2 2 Mah 20 Lecure 2: Cauchy-Euler Equaions Compue [ a()[y y 2 y y 2 ] = ( 2 2 ) ] 2 2 () = 2. () f() y v = 2 () a()[y y 2 y y 2 ] = = 2 ; (2) f() y v 2 = () a() [y y 2 y y 2 ] = 2 = () So ( y p = ) ( + ) 2 =. (4) The general soluion is The Equaion: which can be simplifiied o y =C + C (5) y = C +C 2 2. (6). Basic Informaion a 2 y + b y + c y = 0. (7) Such equaions are called Cauchy-Euler equaions and hey are as easy as equaions of consan coefficiens (In fac, Cauchy-Euler equaions is jus consan-coefficien equaions in disguise, see Noes and Commens). How o ge general soluion Idea: From general heory of 2nd order linear equaions, we know ha as soon as we figure ou wo linearly independen soluions y, y 2, he general soluion is simply C y + C 2 y 2. (8) Recall ha in suing consan-coefficien equaions a y + b y + c y = 0 we obain y, y 2 hrough guess y =e r. Here he idea is similar bu he guess is differen: y = r. Procedure:. Wrie down he characerisic equaion Solve i o ge r, r Three cases: Examples: r r 2, boh real: r = r 2 =r, real. r,2 = α ± iβ.. Wrie down general soluion. a r 2 + (b a) r + c = 0. (9) y = r, y 2 = r2 ; (20) y = r, y 2 = r ln. (2) y = α cos (β ln ), y 2 = α sin (β ln ). (22) Example. Solve 2 y +7y 7 y = 0. (2)
3 Feb., 202 Soluion. Guided by he heory, we only need o find wo linearly independen soluions. The key now is o realize he following propery of r : ( r ) (k) k = C r. Subsiue y = r ino he equaion, we have 0= 2 y +7y 7 y = [r (r )+7r 7] r = 0 r 2 + 6r 7=0 r = 7, r 2 =. (24) Thus he general soluion is y = c 7 + c 2. (25) Example 2. solve 2 y y + 4 y = 0. (26) Soluion. Subsiuing y = r we have r (r ) r + 4=0 r 2 4r +4=0 r = r 2 =2. (27) This is double roo so he general soluion is given by y = c 2 + c 2 2 ln. (28) Example. Solve y y + 5 y = 0. (29) 2 Soluion. Muliply boh sides by 2 : Subsiuing y = r gives 2 y y + 5 y = 0. (0) r (r ) r + 5=0 r 2 2r +5=0 r = +2i, r 2 = 2 i. () The general soluion is hen given by Relaions o consan-coefficien equaions. The formulas r r 2, boh real: r = r 2 =r, real. r,2 = α ± iβ. y = c cos (2 ln )+c 2 sin (2 ln ). (2) y = r, y 2 = r2 ; () y = r, y 2 = r ln. (4) y = α cos (β ln ), y 2 = α sin (β ln ). (5) above looks similar o our heory for linear consan-coeffcien equaions. This similariy becomes more sriking if we inroduce a new variable x=ln : The hree cases become Two disinc roos e rx, e r2x ; One double roo e rx, xe rx ; Complex roos e αx cos β x, e αx sin β x. This is no coincidence! In fac, seing x=ln gives y = d = dx dx d = dx, y = d2 y d 2 = d [ ] dx d 2 y dx dx d = 2 dx 2 2 dx. (6) Subsiuing ino he equaion 0=a 2 y + b y + cy = a d2 y + (b a) + c y. (7) dx2 dx
4 4 Mah 20 Lecure 2: Cauchy-Euler Equaions Thus we have ransformed he Euler-Cauchy equaion ino a consan-coefficien equaion. Furhermore, he auxiliary equaion for his equaion is a r 2 + (b a)r + c = a r (r )+br + c (8) which is exacly he characerisic equaion of he Cauchy-Euler equaion! How o check soluions Noe: Check soluions, especially in he rd case, may involve so much calculaion ha i becomes no worhwhile. Insead, make sure you wrie down he correc characerisic equaion and solve i correcly. See Common Misakes for examples. 2. Things o be Careful/Tricky Issues Fail o see ha he equaion is Cauchy-Euler. (This is he hoes misake in 20!!) Afer finding r,2, wrie e r, e r2 insead of r, r2. (This misake is also very popular.) Iniial value problem.. More Examples Example 4. Solve he following iniial value problem for he Cauchy-Euler equaion Soluion. Subsiuing y = r gives 2 y 4y + 4 y = 0; y()= 2, y () =. (9) r (r ) 4r + 4=0 r 2 5r +4=0 r = 4, r 2 =. (40) Thus he general soluion is given by Using he iniial values, we have Solving his we reach y() =c 4 +c 2. (4) 2 = y() =c +c 2 ; = y ()=4c + c 2. (42) Thus he soluion o he iniial value problem is given by Wha happens if <0? Example 5. (NA; 4.7 5) Solve for <0 Soluion. Le x =. Then x > 0. We have d = dx dx d = dx ; d 2 y d 2 = d ( d d c =, c 2 =. (4) y() = 4 +. (44) y y + 5 = 0. (45) 2 ) = d d ( dx ) = d dx ( dx ) dx d = d2 y dx2. (46) So if we use x insead of as he variable, he equaion (wih unknown y and variable x) reads d 2 y dx 2 x dx + 5 = 0. (47) x2 I is sill Cauchy-Euler, wih a =, b =, c = 5. We wrie down characerisic equaion r 2 2 r +5=0 r,2 = ± 2i. (48)
5 Feb., So he soluion reads Back o : Which can be wrien as y(x) =C x cos (2 ln (x))+c 2 x sin (2 ln x). (49) y()=c ( ) cos(2 ln ( )) +C 2 ( ) sin (2 ln ( )). (50) y()=c cos (2 ln ) +C 2 sin (2 ln ). (5) Remark 6. In fac, we can solve Cauchy-Euler for 0 as r r 2, boh real: r = r 2 =r, real. r,2 = α ± iβ. Wha if i s no? Example 7. (4.7 2; 4.7 2) Solve y = r, y 2 = r2 ; (52) y = r, y 2 = r ln. (5) y = α cos (β ln ), y 2 = α sin (β ln ). (54) ( 2) 2 y 7( 2) y + 7 y = 0, >2. (55) Soluion. I is clear ha we should inroduce x = 2. Now chain rule gives The equaion becomes This can be easily solved: Back o : Nonhomogeneous problem. d = dx ; d 2 y d 2 = d2 y dx2. (56) x 2 d 2 y 7x +7 y =0, x > 0. (57) dx2 dx y(x)=c x+c 2 x 7. (58) y()=c ( 2)+C 2 ( 2) 7. (59) There are wo ways o aack nonhomogeneous problem for Cauchy-Euler equaions. Examples.. Inroduce x = ln, ransform i o consan-coefficien case, hen apply undeermined coefficiens, or variaion of parameers; 2. Apply variaion of parameers direcly. Example 8. Solve Soluion (x = ln ). We know ha seing x=ln ransforms 2 y 4y + 4 y = 2. (60) a 2 y +by + c y o a d2 y + (b a) + cy. (6) dx2 dx So he equaion for x is (x=ln = e x ): d 2 y 5 dx2 dx + 4 y = e2x. (62)
6 6 Mah 20 Lecure 2: Cauchy-Euler Equaions We see ha his equaion is eligible for undeermined coefficiens (keep in mind ha whenever undeermined coefficiens applys, i is more efficien han variaion of parameers). To solve i we firs ge y, y 2 by solving which gives Now guess d 2 y 5 dx2 dx + 4 y = 0 (6) r = 4, r 2 = ; y = e 4x, y 2 =e x. (64) y p = A x s e 2x. (65) We have s = 0 because 2 does no appear in he roo lis r = 4, r 2 =. Subsiue y p =Ae 2x ino he equaion we ge d 2 y dx 2 =4Ae2x, dx =2Ae2x 4 A 0 A+4A= A = 2. (66) So y p = e2x. The general soluion (in x) is hen 2 Back o : (Replace every x by ln ): y = C e 4x + C 2 e x 2 e2x. (67) y()=c 4 +C (68) Soluion 2 (Direc applicaion of variaion of parameers). The equaion is Cauchy Euler so we can solve he homogeneous equaion as follows: So Now calculae: Thus Therefore 2 y 4 y +4 y =0 (69) r (r ) 4r + 4=0 r 2 5r + 4=0 r =4, r 2 =. (70) v = y = 4, y 2 =. (7) a()[y y 2 y y 2 ] = 2 [ 4 4 ] = 6. (72) f() y 2 () a()[y y 2 y y 2 ] = 2 6 = f() y v 2 = () a()[y y 2 y y 2 ] = y p =v y + v 2 y 2 = The general soluion is hen given by Example 9. (4.7 4; 4.7 4) Solve Soluion. Firs solve he homogeneous equaion = 6 2. (7) =. (74) ( 6 ) ( ) = 2 2. (75) y()=c 4 +C (76) 2 z +z + 9z = an ( ln ). (77) 2 z + z +9z = 0. (78)
7 Feb., I is Cauchy-Euler. So firs solve he characerisic equaion: So we have r (r )+r+9=0 r,2 = ±. (79) y = cos ( ln ), y 2 = sin ( ln ). (80) Nex we use variaion of parameers o obain y p. Firs calculae: ( ) a [y y 2 y y 2 ] = [cos 2 ( ln ) sin ( ln ) Now we have ( cos( ln ) ) sin ( ln ) = 2 =. (8) an ( ln ) sin ( ln ) v = d = sin 2 ( ln ) dln. (82) cos ( ln ) Leing x = ln, we compue sin 2 (x) cos (x) dx = dx cos ( x) cos(x)dx = cos 2 (x) cos ( x) dx sin (x) = sin 2 ( x) dsin (x) sin (x) = [ ] dsin ( x) 6 sin ( x) + dsin ( x) sin ( x) + sin ( x) = 6 [ln + sin (x) ln sin ( x) ] sin (x). (8) ] Back o : v = { 6 [ln + sin ( ln ) ln sin ( ln ) ] } sin ( ln ). (84) On he oher hand ( an ( ln )) cos ( ln ) v 2 = d sin ( ln ) = d = sin ( ln )dln Puing hings ogeher we have = cos ( ln ); (85) 9 y p = [ln + sin ( ln ) ln sin ( ln ) ] cos( ln ) 8 sin ( ln ) cos ( ln ) 9 + cos ( ln ) sin ( ln ) 9 = [ln + sin ( ln ) ln sin ( ln ) ] cos( ln ) (86) 8 Thus he soluion is y = C cos ( ln )+C 2 sin ( ln )+ [ln +sin ( ln ) ln sin ( ln ) ] cos ( ln ). (87) 8
8 8 Mah 20 Lecure 2: Cauchy-Euler Equaions Remark 0. For such problems someimes i is more efficien o le x=ln and ransform he equaion (See Noes and Commens). For example for he above problem, if we le x=ln, hen he equaion becomes d 2 y +9 y = an (x). (88) dx2 This can be solved by variaion of parameers, o obain y = C cos (x)+c 2 sin (x)+ [ln +sin ( x) ln sin (x) ] cos ( x). (89) 8 Replace x by ln we ge our soluion. 4. Noes and Commens Why b a? To remember i, jus remember ha our guess is y= r. Subsiuing his ino he equaion we reach which is exacly a r (r )+br + c =0 (90) a r 2 + (b a) r + c = 0. (9) Why do we need > 0? Noe ha, unlike e r, r is singular (meaning: eiher i s infiniy, or is cerain order of derivaive is infiniy) a = 0. More accuraely, when wriing he Cauchy-Euler equaion in sandard form y + b/a y + c/a y = 0. (92) 2 We see ha p() = b/a and q() = c/a are singular a 0. This is an indicaion ha good heories 2 (soluion exiss, soluion is unique, soluion is smooh...) break down when he inerval for conains 0. Therefore o make hings simple we eiher work in > 0 or in < 0, o avoid conaining = 0.
Chapter 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 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 information1. 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
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 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 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 informationWHAT ARE OPTION CONTRACTS?
WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be
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 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 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 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 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 informationOption Put-Call Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
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 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 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 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 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 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 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 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 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 informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The
More information1 HALF-LIFE EQUATIONS
R.L. Hanna Page HALF-LIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of half-lives, and / log / o calculae he age (# ears): age (half-life)
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
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 informationSteps for D.C Analysis of MOSFET Circuits
10/22/2004 Seps for DC Analysis of MOSFET Circuis.doc 1/7 Seps for D.C Analysis of MOSFET Circuis To analyze MOSFET circui wih D.C. sources, we mus follow hese five seps: 1. ASSUME an operaing mode 2.
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 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 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 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 informationMeasuring macroeconomic volatility Applications to export revenue data, 1970-2005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationHFCC Math Lab Intermediate Algebra - 13 SOLVING RATE-TIME-DISTANCE PROBLEMS
HFCC Mah Lab Inemeiae Algeba - 3 SOLVING RATE-TIME-DISTANCE PROBLEMS The vaiables involve in a moion poblem ae isance (), ae (), an ime (). These vaiables ae elae by he equaion, which can be solve fo any
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 informationB-Splines and NURBS Week 5, Lecture 9
CS 430/536 Compuer Graphics I B-Splines an NURBS Wee 5, Lecure 9 Davi Breen, William Regli an Maxim Peysahov Geomeric an Inelligen Compuing Laboraory Deparmen of Compuer Science Drexel Universiy hp://gicl.cs.rexel.eu
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 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 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 informationMaking a Faster Cryptanalytic Time-Memory Trade-Off
Making a Faser Crypanalyic Time-Memory Trade-Off Philippe Oechslin Laboraoire de Securié e de Crypographie (LASEC) Ecole Polyechnique Fédérale de Lausanne Faculé I&C, 1015 Lausanne, Swizerland philippe.oechslin@epfl.ch
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 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 informationVector Autoregressions (VARs): Operational Perspectives
Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101-115. Macroeconomericians
More informationFourier Series & The Fourier Transform
Fourier Series & The Fourier Transform Wha is he Fourier Transform? Fourier Cosine Series for even funcions and Sine Series for odd funcions The coninuous limi: he Fourier ransform (and is inverse) The
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 informationThe Torsion of Thin, Open Sections
EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such
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 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 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 informationPricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
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 information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process,
Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ
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 informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
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 informationC Fast-Dealing Property Trading Game C
If you are already an experienced MONOPOLY dealer and wan a faser game, ry he rules on he back page! AGES 8+ C Fas-Dealing Propery Trading Game C Y Original MONOPOLY Game Rules plus Special Rules for his
More informationStability. Coefficients may change over time. Evolution of the economy Policy changes
Sabiliy Coefficiens may change over ime Evoluion of he economy Policy changes Time Varying Parameers y = α + x β + Coefficiens depend on he ime period If he coefficiens vary randomly and are unpredicable,
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationForecasting, Ordering and Stock- Holding for Erratic Demand
ISF 2002 23 rd o 26 h June 2002 Forecasing, Ordering and Sock- Holding for Erraic Demand Andrew Eaves Lancaser Universiy / Andalus Soluions Limied Inroducion Erraic and slow-moving demand Demand classificaion
More informationC Fast-Dealing Property Trading Game C
AGES 8+ C Fas-Dealing Propery Trading Game C Y Collecor s Ediion Original MONOPOLY Game Rules plus Special Rules for his Ediion. CONTENTS Game board, 6 Collecible okens, 28 Tile Deed cards, 16 Wha he Deuce?
More informationTHE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS
VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
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 informationPRICING and STATIC REPLICATION of FX QUANTO OPTIONS
PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of
More informationCredit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work
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 informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion
More informationSwitching Regulator IC series Capacitor Calculation for Buck converter IC
Swiching Regulaor IC series Capacior Calculaion for Buck converer IC No.14027ECY02 This applicaion noe explains he calculaion of exernal capacior value for buck converer IC circui. Buck converer IIN IDD
More informationA Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets
A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
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 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 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 informationKinematics in 1-D From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.
Chaper 2 Kinemaics in 1-D From Problems and Soluions in Inroducory Mechanics (Draf ersion, Augus 2014) Daid Morin, morin@physics.harard.edu As menioned in he preface, his book should no be hough of as
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 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 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 informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp://www.mecs-press.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecs-press.ne/ijem Opion ricing Under Sochasic Ineres
More information1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z 1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z
o ffix uden abel ere uden ame chool ame isric ame/ ender emale ale onh ay ear ae of irh an eb ar pr ay un ul ug ep c ov ec as ame irs ame lace he uden abel ere ae uden denifier chool se nly rined in he
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 informationNetwork Effects, Pricing Strategies, and Optimal Upgrade Time in Software Provision.
Nework Effecs, Pricing Sraegies, and Opimal Upgrade Time in Sofware Provision. Yi-Nung Yang* Deparmen of Economics Uah Sae Universiy Logan, UT 84322-353 April 3, 995 (curren version Feb, 996) JEL codes:
More informationFull-wave Bridge Rectifier Analysis
Full-wave Brige Recifier Analysis Jahan A. Feuch, Ocober, 00 his aer evelos aroximae equais for esigning or analyzing a full-wave brige recifier eak-eecor circui. his circui is commly use in A o D cverers,
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 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 information