Math 3: Matheatical Notatio Purpose: Oe goal i ay course is to properly use the laguage of that subject Differetial Equatios is o differet ad ay ofte see lie a foreig laguage These otatios suarize soe of the ajor cocepts ad ore difficult topics of the uit Typig the helps you lear the aterial while teachig you to properly express atheatics o the coputer Part of your grade is for properly usig atheatical cotet Istructios: Use Word or WordPerfect to recreate the followig docuets Each article is worth poits ad should be eailed to the istructor at jaes@richladedu This is ot a group assiget, each perso eeds to create ad subit their ow otatio Type your ae at the top of each docuet Iclude the title as part of what you type The lies aroud the title are't that iportat, but if you will type ----- at the begiig of a lie ad hit eter, both Word ad WordPerfect will covert it to a lie For expressios or equatios, you should use the equatio editor i Word or WordPerfect The istructor used WordPerfect ad a 4 pt Ties New Roa fot with 75" argis, so they ay ot loo exactly the sae as your docuet If there is a equatio, put both sides of the equatio ito the sae equatio editor box istead of creatig two objects Be sure to use the proper sybols, there are soe istaces where ore tha oe sybol ay loo the sae, but they have differet eaigs ad do't appear the sae as what's o the assiget There are soe useful tips o the website at http://peoplerichladedu/jaes/editor/ If you fail to type your ae o the docuet, you will lose poit Do't type the hits or reiders that appear o the pages These otatios are due before the begiig of class o the day of the exa for that aterial For exaple, otatio 3 is due o the day of the chapter 3 exa ate wor will be accepted but will lose % of its value per class period If I receive your eailed assiget ore tha oe class period before it is due ad you do't receive all poits, the I will eail you bac with thigs to correct so that you ca get all the poits Ay correctios eed to be subitted by the due date ad tie or the origial score will be used
Chapter - Itroductio to Differetial Equatios A liear differetial equatio is oe where all occurreces of the depedet variable ad its derivatives are raised to the first power The order of a differetial equatio is the order of the highest derivative i the equatio f A differetial equatio will have a uique solutio if both f xy, ad y are cotiuous o soe regio Matheatical Models Populatio Dyaics: The rate of populatio growth is proportioal to the total populatio at that tie dp P Radioactive Decay: The rate at which the uclei of a substace decay is proportioal to the uber of uclei reaiig da A Newto's aw of Coolig: The rate at which the teperature of a body chages is proportioal to the differece betwee the teperature of the body ad the surroudig ediu dt T T Cheical Reactios: The rate at which a reactio proceeds is proportioal to the dx X X product of the reaiig cocetratios d q dq Series Circuits: Kirchoff's secod law says R q Et Fallig Bodies: Without air resistace ad a positive upwards directio, dv C d s g g With air resistace (viscous dapig) ad a positive dowward directio, d s ds gv or g dv Slippig Chai: For a chai i otio aroud ad frictioless peg, d x g x Suspeded Cables: If T is the tesio taget to the lowest poit ad W is the portio of the vertical load betwee two poits, the dy dx W T or
Chapter - First-Order Differetial Equatios A first-order DE is separable if it ca be writte i the for dy dx g x h y The stadard for for a liear first-order DE is hoogeeous if f x dy dx P x y f x ad is The solutio to this DE is the su of two solutios y yc yp where yc is the geeral solutio to the hoogeous DE ad y p is the particular solutio to the ohoogeeous DE The procedure ow as variatio of P xdx paraeters leads to a itegratig factor e The error fuctio ad copleetary error fuctios are defied by x t t erf x e ad erfc x e, where erf x erfc x f f For a fuctio z f x, y, the differetial dz dx dy If the fuctio is a costat, the the differetial is A DE of the for x x y M x, y dx N x, y dy a exact differetial equatio if the left had side is a differetial of soe fuctio f xy, If M ad N are cotiuous ad have cotiuous partial derivatives o soe regio, the it is exact if ad oly if N x M y If a DE is exact, the you ca fid the potetial fuctio f xy, by itegratig Mdx ad Ndy ad fidig the uio of all the ters A fuctio is hoogeeous of degree α if it has the property that f tx, ty t f x, y The substitutios y ux or x vy will reduce a hoogeeous equatio to a separable first-order DE is Beroulli's equatio is substitutio u y dy dx P x y f x y ad ca be solved with the
Chapter 3 - Modelig with First-Order Differetial Equatios Kirchoff's aws: et E t be ipressed voltage, it be curret, qt be charge, be iductace, R be resistace, ad C be capacitace Curret ad charge related by it dq Coservatio of Charge ( st law): The su of the currets eterig a ode ust equal the su of the currets exitig a ode Coservatio of Eergy ( d law): The voltages aroud a closed path i a circuit ust su to zero (voltage drops are egative, voltage gais are positive) The voltage drop across a iductor is di d q q C d q dq C The voltage drop across a resistor is dq ir R The voltage drop across a capacitor is The su of the voltage drops is equal to the ipressed voltage R q E t ogistic Equatio: Whe the rate of growth is proportioal to the aout preset ad the aout reaiig before reachig the carryig capacity K, the the resultig DE is ap dp Pa bp ad the solutio is Pt at bp a bp e ota-volterra Predator-Prey Model: If yt xt is the populatio of a predator ad is the populatio of the prey at tie t, the the populatios ca be odeled by the syste of oliear syste of DEs: dx x a by ad dy y d cx
Chapter 4 - Higher-Order Differetial Equatios Superpositio Priciple - Hoogeeous Equatios: A liear cobiatio of solutios to a hoogeeous DE is also a solutio This eas that costat ultiples of a solutio to a hoogeeous DE are also solutios ad the trivial solutio y is always a solutio to a hoogeeous DE A set of fuctios is liearly depedet if there is soe liear cobiatio of the fuctios that is zero for every x i the iterval A set of solutios is liearly idepedet if ad oly if the Wrosia is ot zero for every x i soe iterval A set of liearly idepedet solutios to a hoogeeous DE is set to be a fudaetal set of solutios ad there is always a fudaetal set for a hoogeeous DE y y y y y y The Wrosia is defied by Wy, y,, y y y y Ay fuctio free of arbitrary paraeters that satisfies a ohoogeeous DE is a particular solutio, y The copleetary fuctio, y, is the geeral solutio to the p associated hoogeeous DE The geeral solutio to a ohoogeeous equatio is y y y c p Reductio of Order: If i stadard for y x y P x y Q x y c is a solutio to a secod-order liear hoogeeous DE, the a secod solutio is Pxdx y x y x dx, where e is the itegratig factor fro chapter y x Hoogeeous iear Equatios with Costat Coefficiets: The auxiliary equatio is fored by covertig the DE ito a polyoial fuctio For exaple, 5 4 3y y 78y 34y 99y 6y would have a auxiliary equatio of 5 4 3 3 78 34 996 You fid the solutios to the auxiliary equatio, which i this case are with ultiplicity, 3, ad 3i Fro each of the roots, we for a liear idepedet cobiatio of ters ivolvig e
x 3 4 5 x x 3 x Thus y ce c xe c e e c cos3xc si 3x Two coo DEs y y ad y y have solutios of x x y ccos xcsi x ad y ce ce respectively The solutios to y y ca also be writte as y ccosh x csih x Method of Udeteried Coefficiets - Superpositio Approach: This ethod is useful whe the coefficiets of the DE are costats ad the iput fuctio is coprised of sus or products of costat, polyoial, expoetial, or trigooetric (sie ad cosie) fuctios You ae guesses about the particular solutios based o the for of the iput ad the equate coefficiets Method of Udeteried Coefficiets - Aihilator Approach: is a aihilator of a fuctio if it has costat coefficiets ad f x I each case below, is a whole uber less tha Use D to aihilate fuctios of the for x ax Use D to aihilate fuctios of the for x e Use ax D D to aihilate fuctios of the for or x e cos x ax x e si x Variatio of Paraeters: Variatio of paraeters ca be used whe the coefficiets of the DE are ot costats It ivolves the Wrosia, W, ad two fuctios y f x yf u x ad u that are itegrated to fid u ad u The particular W W solutio is the y u y u y p d y Cauchy-Euler Equatio: A liear differetial equatio coposed of ters ax, a where the factors are costat, ca be solved by tryig y x Treat it lie the auxiliary equatio, except use l x istead of x For exaple, if the solutios are l x 3i, the y e ccos3l x csi 3l x, which siplifies to y x ccos3l x csi 3l x x dx
Chapter 5 - Modelig with Higher-Order Differetial Equatios Free Udaped Motio: d x d x x ca be writte as x where ad has a solutio of x t c costc sit Free Daped Motio: x d x dx where ad et d If, the overdaped syste has the solutio ca be writte as d x dx x d xt e t ce ce t If, the critically daped syste has the solutio x t e c c t d If d, the uderdaped syste has the solutio t cos si x t e c t d c t d Drive Motio: I drive otio, a exteral force f t is applied to the syste ad d x dx the DE is x F t where F t Use the ethod of udeteried coefficiets or variatio of paraeters to solve the ohoogeeous equatio Series Circuit Aalogue: The DE f t d q dq R C q E t is overdaped, critically daped, or uderdaped depedig o the value of the discriiat R 4 C d y Deflectio of a Bea: Deflectio satisfies the DE EI w x where EI is the flexural rigidity ad w x is the load per uit legth 4 4 dx
Chapter 6 - Series Solutios of iear Equatios If x x is a ordiary poit, the a power series cetered at is y c xx Method of Frobeius: If solutio of the for y c xx r x x x is a regular sigular poit the there exists at least oe r y xx c xx which siplifies to, where r is a costat to be deteried Bessel's Equatio of Order v: The solutio is xyxy x v y v y c J x c Y x y c J x c J x v If v is iteger the the solutio is as log as v is ot a iteger v v Techically, this is a solutio to ay Bessel's equatio, but we prefer J ad whe v is ot a iteger Modified Bessel Equatio: The solutio to xy xy x v y is y ci x c K x If v is v J v v ot a iteger, the you ca use I ad lie we did with the Bessel Equatio egedre's Equatio of Order : v I v x y xy y If is a o- P x egative iteger, the is the solutio ad soe of the solutios are P x, P x x, P x 3x, P x 5x 3 3x 3, ad v 4 35 3 3 If is ot a o-egative iteger, the the solutio is a P x x x 4 8 ifiite series
Chapter 7 - The aplace Trasfor et f be a fuctio defied for t The aplace trasfor of f t is st F s f t e f t, provided this itegral coverges aplace trasfor of a derivative: f t s F s s f s f f First Traslatio Theore: at e f t Fs a Uit Step Fuctio: Also ow as the Heaviside fuctio, it is useful for creatig, t a piecewise fuctios U ta, t a as Secod Traslatio Theore: If the f ta U ta e F s a Derivatives of Trasfors: d t f t Fs ds Covolutio: Covolutios, defied by f g g f t f g f g t d are coutative,, ad the aplace trasfor of a covolutio is the product of the aplace t F s f d trasfors, f g f t g t F s G s If you let g t, the the trasfor of a itegral is s Trasfor of a Periodic Fuctio: period is T T st f t e f t, where the st e Dirac Delta Fuctio: tt li a tt is whe t t ad otherwise a tt st ad tt e
Chapter 8 - Systes of iear First-Order DEs Eigevalues ad Eigevectors: If X AX is a hoogeeous liear first-order syste, the the polyoial equatio det A I is the characteristic equatio ad its solutios are the eigevalues We wat to write a solutio as X Ke t where K is the associated eigevector The geeral solutio to a hoogeous liear syste is XcK e c K e c K e t t t Ke t If your solutios correspod to a coplex eigevalue i, the ad are both solutios Ke t For a ohoogeeous syste, the geeral solutio becoes X Xc Xp ad the ethod of udeteried coefficiets or variatio of paraeters ca be used to fid the particular solutio Matrix Expoetials: For a hoogeeous syste, we ca defie a atrix expoetial t e A so that X e At C is a solutio to X AX For ay square atrix of size, e 3 3 At t IAtA! A 3!, which ca be writte as e A! e At is a At t t fudaetal atrix For ohoogeeous systes, t X AX F t, the geeral solutio is At At As s X Xc Xp e Ce e F s ds I practice, e A ca be foud fro e At by substitutig t s t
Chapter 9 - Nuerical Solutios of Ordiary DEs Euler's Method: I chapter (ad i Calculus II), we had Euler's Method, where y y h f x, y Iproved Euler's Method: This ethod estiates the ext y value i the sequece * usig Euler's ethod, y y h f x, y, ad the uses that estiate i a * idpoit forula to fid the ext y used y y h f x, y f x, y Ruge-Kutta Methods: These are geeralizatios of Euler's ethod where the slope f x, y is replaced by a weighted average of the slopes o the iterval x x x That is, y y h w w w that they agree with a Taylor series of order, where the weights w are chose so RK: The first-order Ruge-Kutta ethod is actually Euler's ethod Choose f x, y ad w to get y y h f x, y RK: The secod-order Ruge-Kutta ethod chooses values f x y,,,, f x h, y h, ad w w to get the iproved Euler's ethod where y y h f x y f x h y h RK4: et w w ad w w Choose f x y, 4 6 3 3,,, 3 f x h, y h, ad, f x h y h f x h y h 4 3