DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations"

Transcription

1 DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows he funcions of he SECOND, APLHA, and GREEN DIAMOND keys. Thus we will simply say display he Y=edior assuming ha he reader will firs press he DIAMOND key and hen F. We have use bold face upper case leers o refer o he calculaor s commands or keys. When we say You can access he command and from Mah 8 8, we mean firs press MATH. (which is nd hen 5) and hen press 8 wice. Noe ha mos menus have several submenus, which in urn may have many opions. To display and selec an opion we may need o use he curser keys. Mos buil-in funcions can be found by pressing CATALOG followed by he firs leer of he desired command. One can hen selec he command by scrolling down, if necessary, using he cursor key. To clear he home screen, use F 8. To go back o he home screen, use HOME or ESC or QUIT. To obain an approximae value (in decimals) use he GREEN DIAMOND followed by ENTER. To draw a graph use GREEN DIAMOND followed by F3. A. Direcion Fields and Graphs of Differenial Equaions Example : Draw he direcion filed of he differenial equaion y' = x y Soluion:. Press he MODE key and from he GRAPH mode selec 6: DIFF EQUATIONS.. Display he Y= edior and ener your differenial equaion. Use for independen variable and y for y. For he prime noaion for he derivaive use ND =. Draw he graph. The figures below show he above seps. Example : Draw he graph of he soluion of y' = x y ha passes hrough (,-) Soluion: Ener your differenial equaion as in Example.. Use he cursor key o move up o he iniial value of and press F3. Then ype and ENTER.. Type for he y-value in he line y i =. Draw he graph. The firs figure below shows he inpus while he second shows he graph. Remarks:. To draw he graph of he soluion wihou he direcion field, from he Y= edior press F 9 o display he GRAPH FORMAT. From he Fields opion (he las row) use he righ cursor key o choose 3:FLDOFF and press ENTER. Now draw he graph.

2 . You can obain soluions passing hrough oher poins by simply changing he iniial condiions while you are in he direcion field/graph screen. To do his press F8 (which is ND F3.) and ype he iniial condiion for and ENTER and hen ype he iniial value for y and ENTER. 3. There are several syles for he graph of he soluion hrough a given poin. From he Y= edior screen press F6 (which is nd F) and choose any of he syles and see wha happens. The figures below show GRAPH FORMAT and he graph of he equaion y' = x ywih differen iniial values using F8 (from he graph display). Example 3: Draw he direcion filed for he sysem of differenial equaions dx = 5x+ 3y d dy = 4x 3y d Soluion: In he Y= edior use y for x and y for y. From he Fields opion of he graph forma, choose :DIRFLD. (See Remark above.) The firs wo figures show he inpus and he direcion field, respecively. The las wo figures show a graph of an iniial value problem for sysems of equaion. (Wrie he IVP.) Example 4: Draw he direcion filed for y'' + 3 y' y = x. Soluion. TI-89 draws direcion fields only for firs order and sysems of firs order differenial equaions. Thus we need o conver his second order equaion in o sysems of firs order equaions. As before we use for he independen variable and for y. We le y = y ' Then he given equaion is equivalen o he sysem y dy = y d dy = + y 3 y d Proceed as in Example 3. The main poin of his example is ha we can use his echnique for higher order differenial equaions. (See Secion D below) B. Solving Firs and Second Order Differenial Equaions The command for solving firs and second differenial equaions is desolve( which can be accessed by F3 C. The forma for his command is desolve(he differenial eqn, independen variable, dependen variable) Example 5: Solve a) y' = y+ x b) y'' + 5 y' 6y = 0 Soluion: In he inpu line ener desolve( y' = y+ x, x, y ) ENTER. The figures ell he sory!

3 x Noe ha TI 89 is giving you he general soluion for a) as y e x x. sands for an arbirary parameer (ha we usually wrie as c or.) x Example 6: Solve he iniial value problem (IVP) y'' + y' 3y =e, y (0) = and y '(0) = c x Soluion: In he inpu line ener desolve( y'' + y' 3y = e and y (0) = and y '(0) =, x, y) ENTER You can access and from Mah ( which is nd 5) 8 8. You could also ype i. Use ALPHA (-) for space. x Example 7: Draw he graph of he soluion of he IVP y'' + y' + 5y =e, y (0) = and y '(0) = Soluion: Firs solve he IVP as in Example 6.Then from he inpu line use F4 o Define y(x) as he soluion. You can ge he soluion by using he upward cursor key and pressing ENTER. Make sure o delee y. This will auomaically ener he soluion as y on he Y=edior. Here is he soluion and he graph. (We have used [-5,8] by [-0,30] for he graph window.) Make sure ha he MODE is on Funcions no on Diff Equaions. C. Euler and Runge-Kua Mehods The TI 89 can generae numerical soluions using he Euler and Runge-Kua mehods. The command for his is BldDaa name. Here BldDaa is he command for building he able of values and name is he name of he daa. We show he deails in he following example. Example 8: Consider he IVP: y' = x y, y (0) = a) Use Euler s mehod o consruc a numerical soluion for he IVP. b) Use Runge-Kua mehod o consruc a numerical soluion for he IVP c) Solve he IVP d) Consruc a able o compare he wo numerical mehods and he exac soluion. Soluions: a) Here are he seps o build he daa for he numerical soluion using Euler s mehod.. Ener he differenial equaion in he Y= edior and delee any oher equaions.. From he graph forma, selec EULER for Soluions Mehod and FLDOFF for Fields. 3. Press HOME hen CATALOG and b. Selec BldDaa and ENTER. Type eu for he name and ENTER. (You could also ype blddaa eu on he inpu line afer your press HOME) 4. Open eu using APPS 6 and hen selec eu for Variable. b) For he Runge-Kua mehod from he graph forma, selec RK for Soluions Mehod. Press HOME and ype blddaa rk and ENTER.

4 c) Solve he IVP using desolve( and Define y(x) as he soluion.(see Example 7.) d) To compare he wo mehods and he exac soluion, follow hese seps.. Use APPS 6 3 o creae a new daa, call i comp.. Press F4 and ype eu[] and ENTER.(eu[] refers o he firs column of he daa called eu.) 3. Move o he second column and press F4. Type eu[] and ENTER. 4. Move o he hird column and press F4. Type rk[] and ENTER. 5. Move o he fourh column and press F4. Type y(c) and ENTER. Here are he figures showing he resuls of he above seps. If your able is differen from ours, change he value of sep in he WINDOWS o 0.. D. Solving Third and Higher Order Differenial Equaions Remark: TI 89 does no solve 3 rd and higher order differenial equaions. To obain he graph of a soluion of hird and higher order equaion, we conver he equaion ino sysems of firs order equaions and draw he graphs.(see Example 4 above.) However, we can uilize he TI 89 capabiliy o solve polynomial equaions wih complex roos o solve linear differenial equaions of higher order wih consan coefficiens. Here are some examples. Example 9: Solve y''' + 3 y '' y' 3y = 0 3 Soluion: The auxiliary equaion is + 3 3= 0 and using he csolve( (which can be accessed by F A ) command for solving equaions wih complex roos, we obain = or = or = 3. Thus he x x 3x general soluion is given by y = ce + ce + c3e. Example 0: Solve y''' + 3 y '' + 8 y' + 6 y = 0. 3 Soluion: The auxiliary equaion is =0 and using csolve( = 0, ) we ge x x x = + 5ior = 5ior =. Thus he general soluion is y = ce e cos(5 x) + c3e + c sin(5x) Example : Solve he IVP y''' y'' 4 y' + 4 y = 0, y (0) = 4, y '(0) =, and y ''(0) = Soluion: The auxiliary equaion here is 4+ 4=0and csolve( 4+ 4= 0, ) yields he x x x soluions = or = or =. Now use F4 o define he general soluion as y = a e + b e + c e. To solve for a, b, c using he iniial condiion, we could ENTER Solve( ( y x= 0) = 4 and ( d( y, x) x= 0) = and ( d( y, x,) x= 0) = 9,a ) For he derivaive, use F3 or nd 8. Here are he seps.

5 Thus he soluion of he IVP is 3 x x y e e e x = +. E. Solving Sysems of Differenial Equaions In Secion A we have discussed how o obain he graph of a soluion of a sysem of differenial equaions. Here we will solve sysems wih consan coefficiens using he heory of eigenvalues and eigenvecors. Example : Solve he sysem of equaions given by X ' = AX where 3 A = Soluion: I is now recommended ha you clear all he single variables you migh have used earlier. F6 ENTER will accomplish his. Here are he relevan seps.. The firs ask will be o ener he marix A. Use APPS 6 3 and Type choose :marix. Use he down cursor key o go o he variable box and ype a for he name of he marix. For boh row and col dimensions ype.(again use he down cursor key afer you yped he inpus.)now ENTER and ype he enries of he marix.(the firs row mus be filled in firs.) Press HOME and CLEAR. Find he eigenvalues of he marix by using Mah 4 9 a) ENTER or by yping eigvl(a) 3. Find he eigenvecors of he marix by using Mah 4 A a) ENTER or by yping eigvc(a) The figures below are he resul of he above seps. The las figure shows he eigenvalues and vecors. Thus he general soluion of he equaion is given by X = c e + c e Remark:. The firs number given by eigvl(a) is he firs eigenvalue which in his case is and second eigenvalue is. The firs column of he eigvc(a) is an eigenvecor corresponding o he firs eigenvalue of a. Noe ha TI 89 is normalizing he vecors, ha is he eigenvecors are uni vecors.. For our purposes and easier noaions, i is convenien o rewrie he eigenvecors wih ineger enries. This is usually possible. One possible mehod is o replace he smalles number in he columns by and divide he oher enries in ha column by he smalles value you jus replaced. Use he command eigvc(a)[j,k] o refer o he j-k enry of he marix eigvc(a).i is clear ha he firs columns are equal hus for he fis eigenvecor we may ake. The second one may no be clear so we replace by. Noe hen ha /-.368 is Thus i is highly recommended ha you compue eigvc(a)[,]/ eigvc(a)[,]. We find ha his is 3. Thus we may ake 3 as he second eigenvecor. Thus he general soluion could also be given by 3. X = c e + c e

6 3. Noe ha we can express he above soluion as e 3 e c. The marix e 3 e b is someimes referred o as he fundamenal marix of he equaion X ' = AX. X = e e c = e e Example 3: Solve X ' = AX,, where X (0) = A 3 = Soluion: As in Example we solve X ' = AX and express he answer in he form given in Remark 3 above. All we have o do now is solve he sysem of equaions e 3e c e e c = 0 = To his end we ener he marix e 3 e ino he calculaor as b. and as d. The we compue he e e command rref(augmen((b =0),d)). The las column of his row reduced echelon form marix gives he soluion for and c. rref and augmen can be accessed from Mah( nd 5) 4 4 and Mah 4 7, respecively. c e F = c = c Example 4: Solve X ' = A X + F, where and 3 A = Soluion: Le b be as in Example and le c. Then he general soluion o he sysem is given by X b c b ( b = + f) d. Ener F as f and execue he command b ( b f) d. For, use F3 or Here are he figures for hese seps. nd 7. Therefore he general soluion is given by 3 e + 3 e 3 e c 4 X = e e c + e + 4 X (0) e 3 = F = A = Example 5: Solve X ' = A X + F,, where and. Soluion: We will use he noaions of Examples 3 and 4. The soluion is hen given by he formula: ( (0)) ( ( ) ( )) X = b b d + b b s f s ds We now need o Define b^(-) and f as funcions of s raher han as funcions of. We will use e and g, for b^(- ) and f, respecively. Here is a parial picure. 0 The soluion is X 3e 5e e e 9e e = + + 3,

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN W will assum ha h radr is familiar wih h calculaor s kyboard and h basic opraions. In paricular w hav assumd ha h radr knows h funcions of

More information

The Transport Equation

The 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 information

Math 201 Lecture 12: Cauchy-Euler Equations

Math 201 Lecture 12: Cauchy-Euler Equations 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

More information

Chapter 7. Response of First-Order RL and RC Circuits

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 information

Fourier Series Solution of the Heat Equation

Fourier Series Solution of the Heat Equation Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

More information

Linear Algebra and TI 89

Linear Algebra and TI 89 Linear Algebra and TI 89 Abdul Hassen and Jay Schiffman This short manual is a quick guide to the use of TI89 for Linear Algebra. We do this in two sections. In the first section, we will go over the editing

More information

4.2 Trigonometric Functions; The Unit Circle

4.2 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 information

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Newton's second law in action

Newton's second law in action Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In

More information

Fourier series. Learning outcomes

Fourier series. Learning outcomes Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series

More information

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed. Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

More information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

More information

AP Calculus BC 2010 Scoring Guidelines

AP 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 information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

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,

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 information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics 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 information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

More information

Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.

Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE. Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,

More information

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (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 information

AP Calculus AB 2013 Scoring Guidelines

AP 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 information

Chabot College Physics Lab RC Circuits Scott Hildreth

Chabot College Physics Lab RC Circuits Scott Hildreth Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard

More information

Differential Equations and Linear Superposition

Differential 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 information

4. The Poisson Distribution

4. The Poisson Distribution Virual Laboraories > 13. The Poisson Process > 1 2 3 4 5 6 7 4. The Poisson Disribuion The Probabiliy Densiy Funcion We have shown ha he k h arrival ime in he Poisson process has he gamma probabiliy densiy

More information

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differential 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 information

Inductance and Transient Circuits

Inductance 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 information

Renewal processes and Poisson process

Renewal processes and Poisson process CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial

More information

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

More information

a. Defining set of equations. Create separate m-file nanme.m containing the set of first order equations.

a. Defining set of equations. Create separate m-file nanme.m containing the set of first order equations. Signals and Sysems. Eperimen. Hins Problems: a) Solving sae equaions using sandard Malab 5. procedures b) Solving ordinary differenial n-h order equaions wihou Symbolic Mah Toolbo (Malab 5.) a.) Solving

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking 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 information

Section 5.1 The Unit Circle

Section 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 information

Revisions to Nonfarm Payroll Employment: 1964 to 2011

Revisions 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 information

Newton s Laws of Motion

Newton 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

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 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 information

Relative velocity in one dimension

Relative velocity in one dimension Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

More information

Use SeDuMi to Solve LP, SDP and SCOP Problems: Remarks and Examples*

Use SeDuMi to Solve LP, SDP and SCOP Problems: Remarks and Examples* Use SeDuMi o Solve LP, SDP and SCOP Problems: Remarks and Examples* * his file was prepared by Wu-Sheng Lu, Dep. of Elecrical and Compuer Engineering, Universiy of Vicoria, and i was revised on December,

More information

13 Solving nonhomogeneous equations: Variation of the constants method

13 Solving nonhomogeneous equations: Variation of the constants method 13 Solving nonhomogeneous equaions: Variaion of he consans meho We are sill solving Ly = f, (1 where L is a linear ifferenial operaor wih consan coefficiens an f is a given funcion Togeher (1 is a linear

More information

AP Calculus AB 2010 Scoring Guidelines

AP 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 information

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009

Full-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 information

Section 7.1 Angles and Their Measure

Section 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 information

Second Order Linear Differential Equations

Second 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 information

Signal Processing and Linear Systems I

Signal 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 information

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendix 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 information

LAB 6: SIMPLE HARMONIC MOTION

LAB 6: SIMPLE HARMONIC MOTION 1 Name Dae Day/Time of Lab Parner(s) Lab TA Objecives LAB 6: SIMPLE HARMONIC MOTION To undersand oscillaion in relaion o equilibrium of conservaive forces To manipulae he independen variables of oscillaion:

More information

Signal Rectification

Signal 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 information

Chapter 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

Chapter 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 information

Economics Honors Exam 2008 Solutions Question 5

Economics 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 information

1 The basic circulation problem

1 The basic circulation problem 2WO08: Graphs and Algorihms Lecure 4 Dae: 26/2/2012 Insrucor: Nikhil Bansal The Circulaion Problem Scribe: Tom Slenders 1 The basic circulaion problem We will consider he max-flow problem again, bu his

More information

Week #9 - The Integral Section 5.1

Week #9 - The Integral Section 5.1 Week #9 - The Inegral Secion 5.1 From Calculus, Single Variable by Hughes-Halle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

( ) in the following way. ( ) < 2

( ) in the following way. ( ) < 2 Sraigh Line Moion - Classwork Consider an obbec moving along a sraigh line eiher horizonally or verically. There are many such obbecs naural and man-made. Wrie down several of hem. Horizonal cars waer

More information

9. Capacitor and Resistor Circuits

9. 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 information

4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 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 information

Theorem Let a, n be relatively prime integers with n > 0. Then the positive integer x is a solution of the congruence. 1 mod n.

Theorem Let a, n be relatively prime integers with n > 0. Then the positive integer x is a solution of the congruence. 1 mod n. 20. Order of an ineger. Primiive roos. Definiion 20.. Le a, n be relaively prime posiive inegers. The leas posiive ineger x such ha a x mod n is called he order of a modulo n. Noaion. ord n a Remark. In

More information

RC Circuit and Time Constant

RC Circuit and Time Constant ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

More information

RC, RL and RLC circuits

RC, 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 information

Steps for D.C Analysis of MOSFET Circuits

Steps 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 information

A Curriculum Module for AP Calculus BC Curriculum Module

A 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 information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE 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 information

Vector Autoregressions (VARs): Operational Perspectives

Vector 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 information

Matrix Analysis of Networks

Matrix Analysis of Networks Marix Analysis of Neworks is edious o analyse large nework using normal equaions. is easier and more convenien o formulae large neworks in marix form. To have a nea form of soluion, i is necessary o know

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,

More information

Part II Converter Dynamics and Control

Part 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 information

The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1

The 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 information

Permutations and Combinations

Permutations 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 information

Basic Assumption: population dynamics of a group controlled by two functions of time

Basic Assumption: population dynamics of a group controlled by two functions of time opulaion Models Basic Assumpion: populaion dynamics of a group conrolled by wo funcions of ime Birh Rae β(, ) = average number of birhs per group member, per uni ime Deah Rae δ(, ) = average number of

More information

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 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 information

Random 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 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 information

Chapter 4: Exponential and Logarithmic Functions

Chapter 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 information

Module 3. R-L & R-C Transients. Version 2 EE IIT, Kharagpur

Module 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 information

Cointegration Analysis of Exchange Rate in Foreign Exchange Market

Cointegration Analysis of Exchange Rate in Foreign Exchange Market Coinegraion Analysis of Exchange Rae in Foreign Exchange Marke Wang Jian, Wang Shu-li School of Economics, Wuhan Universiy of Technology, P.R.China, 430074 Absrac: This paper educed ha he series of exchange

More information

A dynamic probabilistic modeling of railway switches operating states

A dynamic probabilistic modeling of railway switches operating states A dynamic probabilisic modeling of railway swiches operaing saes Faicel Chamroukhi 1, Allou Samé 1, Parice Aknin 1, Marc Anoni 2 1 IFSTTAR, 2 rue de la Bue Vere, 93166 Noisy-le-Grand Cedex, France {chamroukhi,same,aknin}@ifsar.fr

More information

Why Did the Demand for Cash Decrease Recently in Korea?

Why Did the Demand for Cash Decrease Recently in Korea? Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

Two Compartment Body Model and V d Terms by Jeff Stark

Two Compartment Body Model and V d Terms by Jeff Stark Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST 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 information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT 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 information

Stability. Coefficients may change over time. Evolution of the economy Policy changes

Stability. 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 information

MOTION ALONG A STRAIGHT LINE

MOTION 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 information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, 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 information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar

Analogue 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 information

What is a differential equation? y = f (t).

What is a differential equation? y = f (t). Wha is a differenial equaion? A differenial equaion is any equaion conaining one or more derivaives. The simples differenial equaion, herefore, is jus a usual inegraion problem y f (). Commen: The soluion

More information

Chapter 2 Kinematics in One Dimension

Chapter 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 information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal 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 information

Capacitors and inductors

Capacitors 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 information

and Decay Functions f (t) = C(1± r) t / K, for t 0, where

and Decay Functions f (t) = C(1± r) t / K, for t 0, where MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae

More information

11. Properties of alternating currents of LCR-electric circuits

11. Properties of alternating currents of LCR-electric circuits WS. Properies of alernaing currens of L-elecric circuis. Inroducion So-called passive elecric componens, such as ohmic resisors (), capaciors () and inducors (L), are widely used in various areas of science

More information

ACTUARIAL FUNCTIONS 1_05

ACTUARIAL FUNCTIONS 1_05 ACTUARIAL FUNCTIONS _05 User Guide for MS Office 2007 or laer CONTENT Inroducion... 3 2 Insallaion procedure... 3 3 Demo Version and Acivaion... 5 4 Using formulas and synax... 7 5 Using he help... 6 Noaion...

More information

Using RCtime to Measure Resistance

Using 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 buil-in ADC (Analog o Digial Converer)

More information

6.003 Homework #4 Solutions

6.003 Homework #4 Solutions 6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d

More information

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 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

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 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 information

Stochastic Optimal Control Problem for Life Insurance

Stochastic 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 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,

= 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 information

MA261-A Calculus III 2006 Fall Homework 4 Solutions Due 9/29/2006 8:00AM

MA261-A Calculus III 2006 Fall Homework 4 Solutions Due 9/29/2006 8:00AM MA6-A Calculus III 006 Fall Homework 4 Soluions Due 9/9/006 00AM 97 #4 Describe in words he surface 3 A half-lane in he osiive x and y erriory (See Figure in Page 67) 97 # Idenify he surface cos We see

More information

The option pricing framework

The 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 information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 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 information

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity .6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This

More information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 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 information

Laboratory #3 Diode Basics and Applications (I)

Laboratory #3 Diode Basics and Applications (I) Laboraory #3 iode asics and pplicaions (I) I. Objecives 1. Undersand he basic properies of diodes. 2. Undersand he basic properies and operaional principles of some dioderecifier circuis. II. omponens

More information