Mathematical Notation Math Differential Equations

Size: px

Start display at page:

Download "Mathematical Notation Math Differential Equations"

James Barnett
6 years ago
Views:

1 Mathematical Notatio Math 3 - Differetial Equatios Name : Use Word or WordPerfect to recreate the followig documets. Each article is worth poits ad ca be prited ad give to the istructor or ed to the istructor at james@richlad.edu. If you use Microsoft Wors to create the documets, the you must prit it out ad give it to the istructor as he ca t ope those files. Type your ame at the top of each documet. Iclude the title as part of what you type. The lies aroud the title are't that importat, but if you will type at the begiig of a lie ad hit eter, both Word ad WordPerfect will draw a lie across the page for you. For expressios or equatios, you should use the equatio editor i Word or WordPerfect. The documets were created usig a 4 pt Times New Roma fot with stadard " margis. For idividual symbols (:, F, etc), you ca isert symbols. I Word, use "Isert / Symbol" ad choose the Symbol fot. For WordPerfect, use Ctrl-W ad choose the Gree set. For more complex expressios you should use the equatio editor. If there is a equatio, put both sides of the equatio ito the same equatio editor box. You will ru ito trouble i chapter 7, see the otes o the Iteret for help. There are istructios o how to use the equatio editor i a separate documet. Be sure to read through the help it provides. There are some examples at the ed that wal studets through the more difficult problems. You will wat to read the hadout o usig the equatio editor if you have ot used this software before. If you fail to type your ame o the page, you will lose poit. These otatios are due two class periods after we fiish the material for that chapter. See the caledar for exact due dates. Late wor will be accepted but will lose % per class period.

2 Chapter - Itroductio to Differetial Equatios Theorem.: A differetial equatio will have a uique solutio if both f ( xy, ) ad f y are cotiuous o some regio. Mathematical Models Populatio Dyamics: The rate of populatio growth is proportioal to total populatio at that time. dp P = Radioactive Decay: The rate at which the uclei of a substace decay is proportioal to the umber of uclei remaiig. da A = Newto's Law of Coolig: The rate at which the temperature a bo chages is proportioal to the differece betwee the temperature of the bo ad the dt surroudig medium. ( T Tm ) = Chemical Reactios: The rate at which a reactio proceeds is proportioal to the dx product of the remaiig cocetratios. = ( α X)( β X) dq dq Series Circuits: Kirchoff's secod law says L + R + q = E () t C Fallig Bodies: Without air resistace ad a positive upwards directio, ds dv m = mg or m mg. With air resistace (viscous dampig) ad a = dv positive egative directio, m mg v or. = ds ds m + = mg Slippig Chai: For a chai i motio aroud ad frictioless peg, d x g x = L 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 dx = W T

3 Chapter - First-Order Differetial Equatios A first-order DE is separable if it ca be writte i the form g h( y) dx = The stadard form for a liear first-order DE is P y f ad is dx + = homogeeous if. The solutio to this DE is the sum of two solutios f ( x ) = y = yc + yp where yc is the geeral solutio to the homogeous DE ad y p is the particular solutio to the ohomogeeous DE. The procedure ow as variatio of parameters leads to a itegratig factor u = e P xdx. The error fuctio ad complemetary error fuctios are defied by x t = erf x e π t ad erfc x = e ad erf + erfc =. x π f f For a fuctio z = f ( x, y), the differetial dz = dx +. If the fuctio is a x y costat, the the differetial is. A DE of the form M ( x y) dx, + N x, y = is a exact differetial equatio if the left had side is a differetial of some fuctio f ( xy, ). If M ad N are cotiuous ad have cotiuous partial M N derivatives o some regio, the it is exact if ad oly if =. If a DE is y x exact, the you ca fid the potetial fuctio f xy, by itegratig Mdx ad N ad fidig the uio of all the terms. A fuctio is homogeeous of degree " if it has the property that f tx, ty = t α f x, y y = ux x = vy. The substitutios or will reduce a homogeous equatio to a separable first-order DE. Beroulli's equatio is substitutio u = y. P x y f x y dx + = ad ca be solved with the

4 Chapter 3 - Modelig with First-Order Differetial Equatios Kirchoff's Laws: Let be impressed voltage, be curret, q t be charge, L be iductace, E () t i( t ) dq R be resistace, ad C be capacitace. Curret ad charge related by i() t =. Coservatio of Charge ( st law): The sum of the currets eterig a ode must equal the sum of the currets exitig a ode. Coservatio of Eergy ( d law): The voltages aroud a closed path i a circuit must sum to zero (voltage drops are egative, voltage gais are positive). di d q The voltage drop across a iductor is L = L. The voltage drop across a dq resistor is ir = R. The voltage drop across a capacitor is q. The sum of the C dq dq voltage drops is equal to the impressed voltage L + R + q = E() t. C Logistic Equatio: Whe the rate of growth is proportioal to the amout preset ad the amout remaiig before reachig the carryig capacity K, the the dp ap resultig DE is P ( a bp ) ad the solutio is P t = at bp + a bp e = () x( t) Lota-Volterra Predator-Prey Model: If is the populatio of a predator ad y() t is the populatio of the prey at time t, the the populatios ca be dx modeled by the system of oliear system of DEs: x( a by) ad = + y( d cx) =

5 Chapter 4 - Higher-Order Differetial Equatios Superpositio Priciple - Homogeeous Equatios: A liear combiatio of solutios to a homogeeous DE is also a solutio. This meas that costat multiples of a solutio to a homogeeous DE are also solutios ad the trivial solutio y = is always a solutio to a homogeeous DE. A set of fuctios is liearly depedet if there is some liear combiatio 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 some iterval. A set of liearly idepedet solutios to a homogeeous DE is set to be a fudametal set of solutios ad there is always a fudametal set for a homogeeous DE. The Wrosia is defied by f f f f f f W( f, f,, f ) = ( ) ( ) ( ) f f f Ay fuctio free of arbitrary parameters that satisfies a ohomogeeous DE is a particular solutio, y. The complemetary fuctio, y, is the geeral solutio to p the associated homogeeous DE. The geeral solutio to a ohomogeeous equatio is y = yc + yp Reductio of Order: If DE i stadard form y c is a solutio to a secod-order liear homogeeous y + P x y + Q x y = Pxdx e y = y dx y, the a secod solutio is Homogeeous Liear Equatios with Costat Coefficiets: The auxiliary equatio is formed by covertig the DE ito a polyomial fuctio. For example, ( 5) ( 4) 3y y + 78y 34y + 99y 6y = would have a auxiliary equatio of m m + 78m 34m + 99m 6 =. Fid the solutios to the auxiliary

6 equatio, which i this case are m = with multiplicity, m = /3, ad m= ± 3i. From each of the roots, we form a liear idepedet combiatio of x x x 3 x terms ivolvig e. Thus y = ce + c xe + c e + e c cos3x+ c si 3x. Two commo DEs y + y = ad y y = have solutios of x x y = ccos x+ csi x ad y = ce + ce respectively. The solutios to y y = ca also be writte as y = ccosh x+ csih x. Method of Udetermied Coefficiets - Superpositio Approach: This method is useful whe the coefficiets of the DE are costats ad the iput fuctio is comprised of sums or products of costat, polyomial, expoetial, or trigoometric (sie ad cosie) fuctios. You mae guesses about the particular solutios based o the form of the iput ad the equate coefficiets. Method of Udetermied Coefficiets - Aihilator Approach: L is a aihilator of a fuctio if it has costat coefficiets ad L f x =. Use ( ) to aihilate fuctios of the form. Use D α to aihilate fuctios of the x x form. Use D αd α β + + to aihilate fuctios of the form α x α x x e cos β x or x e si β x. I each case, the is a whole umber less tha. x e α Variatio of Parameters: Variatio of parameters ca be used whe the coefficiets of the DE are ot costats. It ivolves the Wrosia ad two y f yf fuctios u = ad u = that are itegrated to fid u ad W W. The particular solutio is the y = u y + u y. u p Cauchy-Euler Equatio: A liear differetial equatio composed of terms d y m ax, where the a factors are costat, ca be solved by tryig. y = x Treat lie the auxiliary equatio, except use l x istead of x. For example, if the lx solutios are m= ± 3i, the y = e ccos( 3l x) + csi ( 3l x), which simplifies to y = x ccos ( 3l x) + csi ( 3l x). D

7 Chapter 5 - Modelig with Higher-Order Differetial Equatios d x d x Free Udamped Motio: m = x ca be writte as + ω x = where ω = ad has a solutio of x( t) = ccosωt+ csiωt. m d x dx Free Damped Motio: m = x β ca be writte as d x dx β + λ + ω x = where λ = ad ω =. If λ ω >, the system is m m t overdamped ad t t x t e λ ce λ ω c e λ ω = +. If λ ω =, the system is () ( ) λt critically damped ad the solutio is x t = e c + c t. If λ ω <, the system is uderdamped ad the solutio is λt x() t = e ccos( t ω λ ) + csi ( t ω λ ). Drive Motio: I drive motio, a exteral force f ( t) is applied to the system d x dx f t ad the DE is + λ + ω x = F() t where F. Use the method of () t = m udetermied coefficiets or variatio of parameters to solve the ohomogeeous equatio. dq dq Series Circuit Aalogue: The DE L + R + q = E() t is overdamped, C critically damped, or uderdamped depedig o the value of the discrimiat R 4 L/ C. Deflectio of a Beam: Deflectio satisfies the DE flexural rigidity ad w EI is the load per uit legth. 4 d y 4 dx = w x where EI is the

8 Chapter 6 - Series Solutios of Liear Equatios If x = x is a ordiary poit, the a power series cetered at x is = y = c x x Method of Frobeius: If least oe solutio of the form = y = c x x. + r x = x is a regular sigular poit the there exists at r y = x x c x x =, where r is a costat to be determied. Bessel's Equatio of Order v: xy+ xy+ x v y= which simplifies to Bessel Fuctios of the First Kid: The two fuctios are + v ( ) x Jv =, which coverges o [,) if v, ad =! Γ + v+ J J v v order v is v ( ) x (, ) Jv =! Γ( v+ ) =, which coverges o. ad are liearly idepedet, so the geeral solutio of Bessel's Equatio of y = c J x + c J x v v where v is ot a iteger. Bessel Fuctios of the Secod Kid: If v is ot a iteger, the Jv x ad cos vπ Jv J v Yv = are liearly idepedet solutios of the Bessel's si vπ y = c J x + c Y x equatio of order v so the geeral solutio ca be writte as v Legedre's Equatio of Order : x y xy y. If P x is + + = P ( x ) = P = x the solutio for order, the some of the solutios are,, ( 3 3 P x = x ), 3 ( P x = x x), ad P4 = ( 35 x 3 x + 3 ). 8 v

9 Chapter 7 - The Laplace Trasform Let f be a fuctio defied for t. The Laplace trasform of f () t is { ()} () { } st f t = e f t, provided this itegral coverges. F s = f t. Laplace trasform of a derivative: ( { ) ()} ( ) f t = s F s s f s f f at First Traslatio Theorem: { e f t } = F( s a) Uit Step Fuctio: Also ow as the Heaviside fuctio, it is useful for, t < a creatig piecewise fuctios. U ( t a) =, t a as Secod Traslatio Theorem: If the f t a U t a = e F s Derivatives of Trasforms: t f () t Covolutio: Covolutios, defied by commutative, f * g = g* f product of the Laplace trasforms, a > { } = ( ) F( s). { } d ds t f * g = f τ g t τ dτ are, ad the Laplace trasform of a covolutio is the { } { } { } f * g = f t g t = F( s) G( s). If { } t F s you let g() t =, the the trasform of a itegral is f ( τ) dτ =. s Trasform of a Periodic Fuctio: period is T. T st f () t = e f t, where the st e { } () Dirac Delta Fuctio: δ t t = limδ a t t is whe t = t ad otherwise. δ t t =. δ ( t t ) = e a st { }

10 Chapter 8 - Systems of Liear First-Order DEs Eigevalues ad Eigevectors: If X = AX is a homogeeous liear first-order system, the the polyomial equatio det A λi = is the characteristic equatio ad it's solutios are the eigevalues. We wat to write a solutio as X= Ke λt where K is the associated eigevector. The geeral solutio to a homogeous solutio is X= ck e λ + c K e λ + + c K e λ t t t Ke λt If your solutios correspod to a complex eigevalue λ = α + βi, the ad Ke λ t are both solutios. For a ohomogeeous system, the geeral solutio becomes X= X + X c p ad the method of udetermied coefficiets or variatio of parameters ca be used to fid the particular solutio. Matrix Expoetials: For a homogeeous system, we ca defie a matrix t expoetial e A so that X= e At C is a solutio to X = AX. For ay square matrix At t t of size, e = I+ At+ A + + A, which ca be writte as!! At t e = A. e At is a fudametal matrix. =! For ohomogeeous systems, t X = AX+ F ( t), the geeral solutio is At At As s X= Xc + Xp = e C+ e e F s ds. I practice, e A ca be foud from t by substitutig t = s. e At

11 Chapter 9 - Numerical 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) Improved Euler's Method: This method estimates the ext y value i the * sequece usig Euler's method, y = y + h + f ( x, y), ad the uses that estimate i a midpoit formula to fid the ext y used. * f ( x, y) + f ( x+, y+ ) y+ = y + h Ruge-Kutta Methods: These are geeralizatios of Euler's method 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 wmm where the weights w are chose so that they agree with a Taylor series of order m. RK: The first-order Ruge-Kutta method is actually Euler's method. Choose = f ( x, y) ad w = to get y = y + h + f ( x, y). RK: The secod-order Ruge-Kutta method chooses values = f x y,, = f ( x + h, y + h), ad w = w = to get the improved Euler's method where y+ = y + h f ( x, y ) + f ( x + h, y + h) RK4: Let w = w4 = ad w = w3 =. Choose = f ( x, y), 6 3 f = x + h, y + h, 3 f = x + h, y + h, ad = f ( x + h, y + h ). 4 3

Math 230: Mathematical Notation

Math 230: Mathematical Notation 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