Math 201 Lecture 12: CauchyEuler Equations


 Shanon McGee
 10 months ago
 Views:
Transcription
1 Mah 20 Lecure 2: CauchyEuler 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: CauchyEuler 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 CauchyEuler equaions and hey are as easy as equaions of consan coefficiens (In fac, CauchyEuler equaions is jus consancoefficien 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 consancoefficien 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 consancoefficien 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 consancoeffcien 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: CauchyEuler Equaions Thus we have ransformed he EulerCauchy equaion ino a consancoefficien 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 CauchyEuler 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 CauchyEuler. (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 CauchyEuler 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 CauchyEuler, 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 CauchyEuler 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 CauchyEuler equaions. Examples.. Inroduce x = ln, ransform i o consancoefficien 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: CauchyEuler 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 CauchyEuler. 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: CauchyEuler 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 CauchyEuler 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 FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder 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 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 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 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 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 information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
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 kvalue for he middle erm, divide
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(ResisorCapacior 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 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 1D. 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 informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou 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 informationEntropy: From the Boltzmann equation to the Maxwell Boltzmann distribution
Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are
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 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 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 discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More informationChapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr
Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han 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 informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationRotational Inertia of a Point Mass
Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha
More informationFullwave rectification, bulk capacitor calculations Chris Basso January 2009
ullwave 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 informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
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 informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More informationSignal Processing and Linear Systems I
Sanford Universiy Summer 214215 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 1415, Gibbons
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67  FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1  TRANSIENTS
EDEXEL NAIONAL ERIFIAE/DIPLOMA UNI 67  FURHER ELERIAL PRINIPLE NQF LEEL 3 OUOME 2 UORIAL 1  RANIEN Uni conen 2 Undersand he ransien behaviour of resisorcapacior (R) and resisorinducor (RL) D circuis
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 informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
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 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 informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
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 imeseries smoohing forecasing mehods. Various models are discussed,
More information4.2 Trigonometric Functions; The Unit Circle
4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.
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 informationSection 7.1 Angles and Their Measure
Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed
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 informationVariance Swap. by Fabrice Douglas Rouah
Variance wap by Fabrice Douglas Rouah www.frouah.com www.volopa.com In his Noe we presen a deailed derivaion of he fair value of variance ha is used in pricing a variance swap. We describe he approach
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 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 informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationSolution of a differential equation of the second order by the method of NIGAM
Tire : Résoluion d'une équaion différenielle du second[...] Dae : 16/02/2011 Page : 1/6 Soluion of a differenial equaion of he second order by he mehod of NIGAM Summarized: We presen in his documen, a
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationHouse Price Index (HPI)
House Price Index (HPI) The price index of second hand houses in Colombia (HPI), regisers annually and quarerly he evoluion of prices of his ype of dwelling. The calculaion is based on he repeaed sales
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
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 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 informationSection 5.1 The Unit Circle
Secion 5.1 The Uni Circle The Uni Circle EXAMPLE: Show ha he poin, ) is on he uni circle. Soluion: We need o show ha his poin saisfies he equaion of he uni circle, ha is, x +y 1. Since ) ) + 9 + 9 1 P
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 ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationPhysics 111 Fall 2007 Electric Currents and DC Circuits
Physics 111 Fall 007 Elecric Currens and DC Circuis 1 Wha is he average curren when all he sodium channels on a 100 µm pach of muscle membrane open ogeher for 1 ms? Assume a densiy of 0 sodium channels
More informationRevisions to Nonfarm Payroll Employment: 1964 to 2011
Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm
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 informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase 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 informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
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 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 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, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationHFCC Math Lab Intermediate Algebra  13 SOLVING RATETIMEDISTANCE PROBLEMS
HFCC Mah Lab Inemeiae Algeba  3 SOLVING RATETIMEDISTANCE 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 informationDensity Dependence. births are a decreasing function of density b(n) and deaths are an increasing function of density d(n).
FW 662 Densiydependen populaion models In he previous lecure we considered densiy independen populaion models ha assumed ha birh and deah raes were consan and no a funcion of populaion size. Longerm
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 informationTorsion of Closed Thin Wall (CTW) Sections
9 orsion of losed hin Wall (W) Secions 9 1 Lecure 9: ORSION OF LOSED HIN WALL (W) SEIONS ALE OF ONENS Page 9.1 Inroducion..................... 9 3 9.2 losed W Secions.................. 9 3 9.3 Examples......................
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationIssues Using OLS with Time Series Data. Time series data NOT randomly sampled in same way as cross sectional each obs not i.i.d
These noes largely concern auocorrelaion Issues Using OLS wih Time Series Daa Recall main poins from Chaper 10: Time series daa NOT randomly sampled in same way as cross secional each obs no i.i.d Why?
More informationA Mathematical Description of MOSFET Behavior
10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical
More informationBSplines and NURBS Week 5, Lecture 9
CS 430/536 Compuer Graphics I BSplines 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 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 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 information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS 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 cuingedge
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
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 noforprofi membership associaion whose mission is o connec sudens o college success and
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 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), 101115. Macroeconomericians
More informationMaking a Faster Cryptanalytic TimeMemory TradeOff
Making a Faser Crypanalyic TimeMemory TradeOff 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 informationMOTION ALONG A STRAIGHT LINE
Chaper 2: MOTION ALONG A STRAIGHT LINE 1 A paricle moes along he ais from i o f Of he following alues of he iniial and final coordinaes, which resuls in he displacemen wih he larges magniude? A i =4m,
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.unibremen.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 noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy.
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen 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 bipolar
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 informationA NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More informationUsing RCtime to Measure Resistance
Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a builin ADC (Analog o Digial Converer)
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 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 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 informationC FastDealing Property Trading Game C
AGES 8+ C FasDealing 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 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 slowmoving demand Demand classificaion
More informationSKF Documented Solutions
SKF Documened Soluions Real world savings and we can prove i! How much can SKF save you? Le s do he numbers. The SKF Documened Soluions Program SKF is probably no he firs of your supplier parners o alk
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 information