Linear FirstOrder Equations


 Charity Elaine Walton
 1 years ago
 Views:
Transcription
1 5 Linear FirstOrer Equations Linear firstorer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory) As with the notion of separability, the iea of linearity for firstorer equations can be viewe as a simple generalization of the notion of irect integrability, an a relatively straightforwar (though, perhaps, not so intuitively obvious) metho will allow us to put any firstorer linear equation into a form that can be relatively easily integrate We will erive this metho in a short while (after, of course, escribing just what it means for a firstorer equation to be linear ) By the way, the criteria given here for a ifferential equation being linear will be extene later to higherorer ifferential equations, an a rather extensive theory will be evelope to hanle linear ifferential equations of any orer That theory is not neee here; in fact, it woul be of very limite value An, to be honest, the basic technics we ll evelop in this chapter are only of limite use when it comes to solving higherorer linear equations However, these basic technics involve an integrating factor, which is something we ll be able to generalize a little bit later (in chapter 7) to help solve much more general firstorer ifferential equations 51 Basic Notions Definitions A firstorer ifferential equation is sai to be linear if an only if it can be written as or, equivalently, as = f (x) p(x)y (51) + p(x)y = f (x) (52) where p(x) an f (x) are known functions of x only Equation (52) is normally consiere to be the stanar form for firstorer linear equations Note that the only appearance of y in a linear equation (other than in the erivative) is in a term where y alone is multiplie by some formula of x If there is any other functions of y appearing in the equation after you ve isolate the erivative, then the equation is not linear 103
2 104 Linear FirstOrer Equations! Example 51: Consier the ifferential equation Solving for the erivative, we get x + 4y x 3 = 0 = x3 4y x = x 2 4 x y, which is with p(x) = 4 x = f (x) p(x)y an f (x) = x 2 So this firstorer ifferential equation is linear Aing 4 / x y to both sies, we then get the equation in stanar form, + 4 x y = x 2, On the other han is not linear because of the y x y2 = x 2 In testing whether a given firstorer ifferential equation is linear, it oes not matter whether you attempt to rewrite the equation as or as = f (x) p(x)y + p(x)y = f (x) If you can put it into either form, the equation is linear You may prefer the first, simply because it is a natural form to look for after solving the equation for the erivative However, because the secon form (the stanar form) is more suite for the methos normally use for solving these equations, more experience workers typically prefer that form! Example 52: Consier the equation x 2 + x 3 [y sin(x)) = 0 Diviing through by x 2 an oing a little multiplication an aition converts the equation to + xy = x sin(x), which is the stanar form for a linear equation So this ifferential equation is linear
3 Basic Notions 105 It is possible for a linear equation + p(x)y = f (x) to also be a type of equation we ve alrea stuie For example, if p(x) = 0 then the equation is = f (x), which is irectly integrable If, instea, f (x) = 0, the equation can be rewritten as = p(x)y, showing that it is separable In aition, you can easily verify that a linear equation is separable if f (x) is any constant multiple of p(x) If a linear equation is also irectly integrable or separable, then it can be solve using methos alrea iscusse Otherwise, a small trick turns out to be very useful Deriving the Trick for Solving Suppose we want to solve some firstorer linear equation + py = f (53) (for brevity, p = p(x) an f = f (x) ) To avoi triviality, let s assume p(x) is not always 0 Whether f (x) vanishes or not will not be relevant The small trick to solving equation (53) comes from the prouct rule for erivatives: If µ an y are two functions of x, then µ [µy = y + µ Rearranging the terms on the right sie, we get [µy = µ + µ y, an the right sie of this equation looks a little like the left sie of equation (53) To get a better match, let s multiply equation (53) by µ, µ + µpy = µf With luck, the left sie of this equation will match the right sie of the last equation for the prouct rule, an we will have This, of course, requires that [µy = µ + µ y = µ (54) + µpy = µf µ = µp
4 106 Linear FirstOrer Equations Assuming this requirement is met, the equations in (54) hol Cutting out the mile of that (an recalling that f an µ are functions of x only), we see that the ifferential equation reuces to [µy = µ(x) f (x) (55) The avantage of having our ifferential equation in this form is that we can actually integrate both sies with respect to x, with the left sie being especially easy since it is just a erivative with respect to x The function µ is calle an integrating factor for the ifferential equation As note in the erivation, it must satisfy µ = µp (56) This is a simple separable ifferential equation for µ (remember, p = p(x) is a known function) Any nonzero solution to this can be use as an integrating factor (the zero solution, µ = 0, woul simplify matters too much!) Applying the approach we learne for separable ifferential equations, we ivie through by µ, integrate, an solve the resulting equation for µ : 1 µ µ = p(x) ln µ = p(x) µ = ±e p(x) Since we only nee one function µ(x) satisfying requirement (56), we can rop both the ± an any arbitrary constant arising from the integration of p(x) This leaves us with a relatively simple formula for our integrating factor; namely, µ(x) = e p(x) (57) where it is unerstoo that we can let the constant of integration be zero 52 Solving FirstOrer Linear Equations As we just erive, the real trick to solving a firstorer linear equation is to reuce it to an easily integrate form via the use of an integrating factor Let me outline a proceure for actually carrying out the necessary steps To illustrate these steps, we will immeiately use them to fin the general solution to the equation from example 51, The Proceure: x + 4y = x 3 1 Get the equation into the stanar form for firstorer linear ifferential equations, + p(x)y = f (x)
5 Solving FirstOrer Linear Equations 107 For our example, we just ivie through by x, obtaining + 4 x y = x 2 As note in example 51, this is the esire form with p(x) = 4 x an f (x) = x 2 2 Compute an integrating factor µ(x) = e p(x) Remember, since we only nee one integrating factor, we can let the constant of integration be zero here For our example, µ(x) = e p(x) = e 4 x = e 4 ln x Applying some basic ientities for the natural logarithm, we can rewrite this last expression in a much more convenient form: µ(x) = e 4 ln x = e ln x4 = x 4 = x 4 3a Multiply the ifferential equation (in stanar form) by the integrating factor, [ µ + p(x)y = f (x) µ + µpy = µf, b an observe that, via the prouct rule an choice of µ, the left sie can be written as the erivative of the prouct of µ an y, µ + µpy = µf, }{{} [µy c an then rewrite the ifferential equation as [µy = µf, For our example, µ = x 4 Multiplying our equation by this an proceeing through the three substeps above, yiels x 4[ + 4 x y = x 2 x 4 + 4x 3 y = x 6 }{{} [x4 y [x 4 y = x 6
6 108 Linear FirstOrer Equations 4 Integrate with respect to x both sies of the last equation obtaine, [µy = µ(x) f (x) µy = µ(x) f (x) Don t forget the arbitrary constant here! Integrating the last equation in our example, [x 4 y = x 6 x 4 y = 1 7 x 7 + c 5 Finally, solve for y by iviing through by µ For our example, [ y = x x 7 + c = 1 7 x 3 + cx 4 It is possible to use the above proceure to erive an explicit formula for computing y from p(x) an f (x) Unfortunately, it is not a particularly simple formula, an those who attempt to memorize it typically make more mistakes than those who simply remember the above proceure So I won t tell you that formula, yet 1! Example 53: Consier e x = ex y, y(0) = 7 Subtracting 3e x y from both sies an then multiplying through by e x ifferential equation into the esire form, puts this linear So p(x) = 3, an our integrating factor is 3y = 20e x µ = µ(x) = e 3 = e 3x Multiplying the ifferential equation by µ an following the rest of the steps in our proceure gives us the following: e 3x[ 3y = 20e x e 3x 3e 3x y = 20e 4x }{{} [e 3x y 1 If you must see this formula, glance ahea to theorem 51 on page 114
7 Solving FirstOrer Linear Equations 109 [e 3x y = 20e 4x [e 3x y = 20e 4x e 3x y = 5e 4x + c y = e 3x [ 5e 4x + c So the general solution to our ifferential equation is Thus, y(x) = 5e x + ce 3x Using this formula for y(x) with the initial conition gives us 7 = y(0) = 5e 0 + ce 3 0 = 5 + c c = = 12, an the solution to the given initialvalue problem is y(x) = 5e x + 12e 3x Let us briefly get back to our requirement for µ = µ(x) being an integrating factor for That requirement was equation (56), + py = f µ = µp Now, in computing this µ, you will often get something like µ(x) = µ 0 (x) where µ 0 (x) is a relatively simple continuous function (eg, µ(x) = sin(x) ) Consequently, on any interval over which the graph of µ 0 (x) never crosses the X axis, µ 0 (x) = µ(x) or µ 0 (x) = µ(x) Either way, µ 0 = [±µ = ± µ = ±µp = µ 0 p So µ 0 also satisfies the requirement for being an integrating factor for the given ifferential equation This means that, if in computing µ you o get something like µ(x) = µ 0 (x) where µ 0 (x) is a relatively simple function, then you can ignore the absolute value brackets an just use µ 0 for your integrating factor
8 110 Linear FirstOrer Equations! Example 54: Consier solving the linear ifferential equation + cot(x)y = x csc(x) This equation is alrea in the esire form In a case like this, it is often a goo iea to see what the equation looks like in terms of sines an cosines, So [ cos(x) + y = sin(x) x sin(x) To fin µ = e p, first observe that, ignoring the constant of integration, cos(x) p(x) = sin(x) = sin(x) = ln sin(x) sin(x) µ = µ(x) = e p(x) = e ln sin(x) = sin(x) As iscusse above, we can just rop the an use sin(x) for the integrating factor Doing so, an stepping through the rest of our proceure, we have [ sin(x) + cos(x) sin(x) y = x sin(x) sin(x) + cos(x)y = x }{{} [sin(x)y [sin(x)y = x sin(x)y = 1 2 x 2 + c 1 y = x2 + c 2 sin(x) 53 On Using Definite Integrals with Linear Equations Integration arises twice in our metho for solving + p(x)y = f (x) It first arises when we integrate p to get the integrating factor, µ(x) = e p(x) It then is neee again when we then integrate both sies of the corresponing equation [µy = µf
9 On Using Definite Integrals with Linear Equations 111 At either point, of course, we coul use efinite integrals instea of inefinite integrals Let s first look at what happens when we integrate both sies of the last equation using efinite integrals Remember, everything is a function of x, so this equation can be written a bit more explicitly as [µ(x)y(x) = µ(x) f (x) As before, to avoi having x represent two ifferent entities, we replace the x s with another variable, say, s, an rewrite our current ifferential equation as [µ(s)y(s) = µ(s) f (s) s Then we pick a convenient lower limit a for our integration an integrate each sie of the above with respect to s from s = a to s = x, But So equation (58) reuces to a a [µ(s)y(s) s = s s [µ(s)y(s) s = µ(s)y(s) x a a µ(s) f (s) s (58) = µ(x)y(x) µ(a)y(a) µ(x)y(x) µ(a)y(a) = a µ(s) f (s) s, Solving this for y(x) yiels y(x) = 1 [ µ(a)y(a) + µ(x) a µ(s) f (s) s (59) This is not a simple enough formula to be worth memorizing (especially since you still have to remember what µ is) Nonetheless, it is a formula worth knowing about for at least two goo reasons: 1 This formula can automatically take into account an initial value y( ) = y 0 All we have to o is to choose the lower limit a to be Then formula (59) tells us that the solution to + py = f with y() = y 0 is y = 1 [ µ( )y 0 + µ(x) µ(s) f (s) s (510) 2 Even if we cannot etermine a relatively nice formula for integral of µf (for a given choice of µ an f ), the value of the integral in formula (59) can, in practice, still be accurately compute for esire values of x using numerical integration routines foun in stanar computer math packages Inee, using any of these packages an formula (59), you coul probably program a computer to accurately compute y(x) for a number of values of x an use these values to prouce a very accurate graph of y
10 112 Linear FirstOrer Equations! Example 55: Consier solving 2xy = 4 with y(0) = 3 The ifferential equation is clearly linear an in the esire form for the first step of our proceure Computing the integrating factor, we fin that, here, µ = e p(x) = e [ 2x = e x2 +c Choosing, as we may, c to be zero, we then get µ(x) = e x2 Plugging this into formula (59) (an choosing a = 0 since we have y(0) = 3 as the initial conition) yiels y(x) = 1 µ(x) = 1 e x2 [ µ(0)y(0) + [ e = e x2 [ e s2 s µ(s) f (s) s e s2 4 s This is the solution to our initialvalue problem The integral, 0 e s2 s, was left unevaluate because no one has yet foun a nice formula for this integral At best, we can hie this integral by using the error function (see page 30), rewriting our formula for y as [ y(x) = e x π erf(x) Still, to fin the value of, say, y(4), we woul have to either numerically approximate the integral in 4 y(4) = e [ e s2 s or look up the value of the error function in y(4) = e 42 [ π erf(4) Either way, a ecent computer math package coul be helpful 0 As alrea note, we coul also use a efinite integral in etermining the integrating factor This means µ woul be given by ( ) µ(x) = exp p(s) s a
11 Integrability an the Existence an Uniqueness of Solutions 113 where a was any appropriate lower limit Naturally, if we ha an initial conition y( ) = y 0, it woul make sense to let a = This woul slightly simplify formula (510) to y = 1 [ y 0 + µ(s) f (s) s (511) µ(x) since (0 ) µ( ) = exp p(s) s = e 0 = 1 In practice, there is little to be gaine in using a efinite integral in the computation of µ unless there is not a reasonable formula for the integral of p Then we are pretty well force into using a efinite integral to compute µ(x) an to computing this integral numerically for each value of x of interest That, in turn, woul pretty well force us to compute y(x) for each x of interest by using numerical computation of formula (510) 54 Integrability an the Existence an Uniqueness of Solutions If you check, you will see that our erivation of the efinite integral formula y(x) = 1 [ y 0 + µ(x) as a solution to the initialvalue problem µ(s) f (s) s with + p(x)y = f (x) with y() = y 0 ( ) µ(x) = exp p(s) s merely require that y be any solution to this problem, an that p an f be sufficiently integrable for the existence of the integrals involving them That is, every solution to this initialvalue problem is given by this one formula Conversely, as long as p an f are sufficiently integrable, you can use elementary calculus to ifferentiate the above efinite integral formula an verify that the y efine by this formula is a solution to the above initialvalue problem (see problem 55) Thus, the above efinite integral formula gives us the one an only solution to the above initialvalue problem, provie p an f are sufficiently integrable Just what is sufficiently integrable? Basically, we want the integrals p(s) s an µ(s) f (s) s to be wellefine, continuous functions of x in whatever interval of interest (α, β) we have (Note that this ensures ( ) µ(x) = exp p(s) s is never zero in this interval) Certainly, p an f will be sufficiently integrable if they are continuous on (α, β) But continuity is not necessary; p an f can have a few iscontinuities
12 114 Linear FirstOrer Equations provie these iscontinuities are not too ba In particular, we can allow the same piecewiseefine functions consiere back in section 24 That (along with theorem 21 on page 35) gives us the following existence an uniqueness theorem for initialvalue problems involving firstorer linear ifferential equations Theorem 51 (existence an uniqueness) Let p an f be functions that are continuous except for, at most, a finite number of finitejump iscontinuities in an interval (α,β) Also let an y 0 be any two numbers with α < < β Then the initialvalue problem + p(x)y = f (x) with y() = y 0 has exactly one solution over the interval (α,β), an that solution is given by y(x) = 1 [ ( ) y 0 + µ(s) f (s) s with µ(x) = exp p(s) s µ(x) Aitional Exercises 51 Determine whether each of the following ifferential equations is or is not linear, an, if it is linear, rewrite the equation in stanar form, + p(x)y = f (x) a x 2 + 3x 2 y = sin(x) b y 2 c e g i xy2 = x + 3x 2 y = sin(x) = 1 + (xy + 3y)2 = 1 + xy + 3y f = 4y + 8 e2x = 0 h + 4y = y3 j x = sin(x) y + cos( x 2) = 827y 52 Using the methos evelope in this chapter, fin the general solution to each of the following firstorer linear ifferential equations: a c + 2y = 6 b + 2y = 20e3x = 4y + 16x e x + 3y 10x 2 = 0 f x 2 2xy = x + 2xy = sin(x)
13 Aitional Exercises 115 g x = x + 3y h cos(x) i x + (5x + 2)y = 20 x j 2 x + sin(x) y = cos2 (x) + y = x 2xe 53 Fin the solution to each of the following initialvalue problems using the methos evelope in this chapter: a b c 3y = 6 with y(0) = 5 3y = 6 with y(0) = 2 + 5y = e 3x with y(0) = y = 20x 2 with y(1) = 10 e x = y + x 2 cos(x) with y ( ) π = 0 2 f (1 + x 2 ) = x [ 3 + 3x 2 y with y(2) = 8 54 Express the answer to each of the following initialvalue problems in terms of efinite integrals: a + 6xy = sin(x) with y(0) = 4 b x 2 + xy = x sin(x) with y(2) = 5 c x y = x 2 e x2 with y(3) = 8 55 Let (α,β) be an interval, an let an y 0 be any two numbers with α < < β Assume p an f are functions continuous at all but, at most, a finite number of points in (α, β), an that each of these iscontinuities is a finitejump iscontinuity Define µ(x) an y(x) by ( ) µ(x) = exp p(s) s an y(x) = 1 [ y 0 + µ(x) µ(s) f (s) s a Compute the first erivatives of µ an y b Verify that y satisfies the initial conition y( ) = y 0 as well as the ifferential equation on (α,β) + p(x)y = f (x)
14
Implicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More information1 Lecture 20: Implicit differentiation
Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation
More information2 ODEs Integrating Factors and Homogeneous Equations
2 ODEs Integrating Factors an Homogeneous Equations We begin with a slightly ifferent type of equation: 2.1 Exact Equations These are ODEs whose general solution can be obtaine by simply integrating both
More informationOrdinary Differential Equations
Orinary Differential Equations Example: Harmonic Oscillator For a perfect Hooke slaw spring,force as a function of isplacement is F = kx Combine with Newton s Secon Law: F = ma with v = a = v = 2 x 2
More informationIntegration by Parts
Integration by Parts 63207 If u an v are functions of, the Prouct Rule says that (uv) = uv +vu Integrate both sies: (uv) = uv = uv + u v + uv = uv vu, vu v u, I ve written u an v as shorthan for u an
More informationApplications of the Wronskian to ordinary linear differential equations
Physics 116C Fall 2011 Applications of the Wronskian to orinary linear ifferential equations Consier a of n continuous functions y i (x) [i = 1,2,3,...,n], each of which is ifferentiable at least n times.
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationMA 2232 Lecture 08  Review of Log and Exponential Functions and Exponential Growth
MA 2232 Lecture 08  Review of Log an Exponential Functions an Exponential Growth Friay, February 2, 2018. Objectives: Review log an exponential functions, their erivative an integration formulas. Exponential
More informationChapter 2. Exponential and Log functions. Contents
Chapter. Exponential an Log functions This material is in Chapter 6 of Anton Calculus. The basic iea here is mainly to a to the list of functions we know about (for calculus) an the ones we will stu all
More informationDerivatives and the Product Rule
Derivatives an the Prouct Rule James K. Peterson Department of Biological Sciences an Department of Mathematical Sciences Clemson University January 28, 2014 Outline Differentiability Simple Derivatives
More informationSection 7.1: Integration by Parts
Section 7.1: Integration by Parts 1. Introuction to Integration Techniques Unlike ifferentiation where there are a large number of rules which allow you (in principle) to ifferentiate any function, the
More informationChapter 6: Integration: partial fractions and improper integrals
Chapter 6: Integration: partial fractions an improper integrals Course S3, 006 07 April 5, 007 These are just summaries of the lecture notes, an few etails are inclue. Most of what we inclue here is to
More informationFirst Order Linear Differential Equations
LECTURE 8 First Orer Linear Differential Equations We now turn our attention to the problem of constructing analytic solutions of ifferential equations; that is to say,solutions that can be epresse in
More informationComputing Derivatives
Chapter 2 Computing Derivatives 2.1 Elementary erivative rules Motivating Questions In this section, we strive to unerstan the ieas generate by the following important questions: What are alternate notations
More informationDefine each term or concept.
Chapter Differentiation Course Number Section.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan
More informationQF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim
QF101: Quantitative Finance September 5, 2017 Week 3: Derivatives Facilitator: Christopher Ting AY 2017/2018 I recoil with ismay an horror at this lamentable plague of functions which o not have erivatives.
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More information4 Exact Equations. F x + F. dy dx = 0
Chapter 1: First Order Differential Equations 4 Exact Equations Discussion: The general solution to a first order equation has 1 arbitrary constant. If we solve for that constant, we can write the general
More informationSome functions and their derivatives
Chapter Some functions an their erivatives. Derivative of x n for integer n Recall, from eqn (.6), for y = f (x), Also recall that, for integer n, Hence, if y = x n then y x = lim δx 0 (a + b) n = a n
More informationThe PressSchechter mass function
The PressSchechter mass function To state the obvious: It is important to relate our theories to what we can observe. We have looke at linear perturbation theory, an we have consiere a simple moel for
More informationx = c of N if the limit of f (x) = L and the righthanded limit lim f ( x)
Limit We say the limit of f () as approaches c equals L an write, lim L. OneSie Limits (Left an RightHane Limits) Suppose a function f is efine near but not necessarily at We say that f has a lefthane
More informationDifferentiability, Computing Derivatives, Trig Review
Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute
More informationTrigonometric Functions
4 Trigonometric Functions So far we have use only algebraic functions as examples when fining erivatives, that is, functions that can be built up by the usual algebraic operations of aition, subtraction,
More informationf(x) f(a) Limit definition of the at a point in slope notation.
Lesson 9: Orinary Derivatives Review Hanout Reference: Brigg s Calculus: Early Transcenentals, Secon Eition Topics: Chapter 3: Derivatives, p. 126235 Definition. Limit Definition of Derivatives at a point
More informationLecture 6: Calculus. In Song Kim. September 7, 2011
Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear
More informationand from it produce the action integral whose variation we set to zero:
Lagrange Multipliers Monay, 6 September 01 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More informationHyperbolic Functions. Notice: this material must not be used as a substitute for attending. the lectures
Hyperbolic Functions Notice: this material must not be use as a substitute for attening the lectures 0. Hyperbolic functions sinh an cosh The hyperbolic functions sinh (pronounce shine ) an cosh are efine
More informationThe Chain Rule. Composition Review. Intuition. = 2(1.5) = 3 times faster than (X)avier.
The Chain Rule In the previous section we ha to use a trig ientity to etermine the erivative of. h(x) = sin(2x). We can view h(x) as the composition of two functions. Let g(x) = 2x an f (x) = sin x. Then
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationq = F If we integrate this equation over all the mass in a star, we have q dm = F (M) F (0)
Astronomy 112: The Physics of Stars Class 4 Notes: Energy an Chemical Balance in Stars In the last class we introuce the iea of hyrostatic balance in stars, an showe that we coul use this concept to erive
More informationQubit channels that achieve capacity with two states
Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March
More informationChapter 2 The Derivative Business Calculus 155
Chapter The Derivative Business Calculus 155 Section 11: Implicit Differentiation an Relate Rates In our work up until now, the functions we neee to ifferentiate were either given explicitly, x such as
More informationG j dq i + G j. q i. = a jt. and
Lagrange Multipliers Wenesay, 8 September 011 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine
More information18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.eu 8.0 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.eu/terms. Lecture 8.0 Fall 2006 Unit
More information11.7. Implicit Differentiation. Introduction. Prerequisites. Learning Outcomes
Implicit Differentiation 11.7 Introuction This Section introuces implicit ifferentiation which is use to ifferentiate functions expresse in implicit form (where the variables are foun together). Examples
More informationAntiderivatives Introduction
Antierivatives 40. Introuction So far much of the term has been spent fining erivatives or rates of change. But in some circumstances we alreay know the rate of change an we wish to etermine the original
More informationIntroduction to Markov Processes
Introuction to Markov Processes Connexions moule m44014 Zzis law Gustav) Meglicki, Jr Office of the VP for Information Technology Iniana University RCS: Section2.tex,v 1.24 2012/12/21 18:03:08 gustav
More informationd dx [xn ] = nx n 1. (1) dy dx = 4x4 1 = 4x 3. Theorem 1.3 (Derivative of a constant function). If f(x) = k and k is a constant, then f (x) = 0.
Calculus refresher Disclaimer: I claim no original content on this ocument, which is mostly a summaryrewrite of what any stanar college calculus book offers. (Here I ve use Calculus by Dennis Zill.) I
More informationChain rules via multiplication
Chain rules via multiplication Bro. Davi E. Brown, BYU Iaho Dept. of Mathematics. All rights reserve. Version 0.44, of June 16, 2014 Answer to Exercise 2.1 correcte, minor eits mae, numbering of exercises
More information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk  Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking threeimensional potentials in the next chapter, we shall in chapter 4 of this
More informationAn Optimal Algorithm for Bandit and ZeroOrder Convex Optimization with TwoPoint Feedback
Journal of Machine Learning Research 8 07)  Submitte /6; Publishe 5/7 An Optimal Algorithm for Banit an ZeroOrer Convex Optimization with wopoint Feeback Oha Shamir Department of Computer Science an
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationSection 2.1 The Derivative and the Tangent Line Problem
Chapter 2 Differentiation Course Number Section 2.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan
More informationBy writing (1) as y (x 5 1). (x 5 1), we can find the derivative using the Product Rule: y (x 5 1) 2. we know this from (2)
3.5 Chain Rule 149 3.5 Chain Rule Introuction As iscusse in Section 3.2, the Power Rule is vali for all real number exponents n. In this section we see that a similar rule hols for the erivative of a power
More informationNotes on Lie Groups, Lie algebras, and the Exponentiation Map Mitchell Faulk
Notes on Lie Groups, Lie algebras, an the Exponentiation Map Mitchell Faulk 1. Preliminaries. In these notes, we concern ourselves with special objects calle matrix Lie groups an their corresponing Lie
More informationCenter of Gravity and Center of Mass
Center of Gravity an Center of Mass 1 Introuction. Center of mass an center of gravity closely parallel each other: they both work the same way. Center of mass is the more important, but center of gravity
More informationWhy Bernstein Polynomials Are Better: FuzzyInspired Justification
Why Bernstein Polynomials Are Better: FuzzyInspire Justification Jaime Nava 1, Olga Kosheleva 2, an Vlaik Kreinovich 3 1,3 Department of Computer Science 2 Department of Teacher Eucation University of
More information1 The Derivative of ln(x)
Monay, December 3, 2007 The Derivative of ln() 1 The Derivative of ln() The first term or semester of most calculus courses will inclue the it efinition of the erivative an will work out, long han, a number
More informationSeparable FirstOrder Equations
4 Separable FirstOrder Equations As we will see below, the notion of a differential equation being separable is a natural generalization of the notion of a firstorder differential equation being directly
More informationModule FP2. Further Pure 2. Cambridge University Press Further Pure 2 and 3 Hugh Neill and Douglas Quadling Excerpt More information
5548993  Further Pure an 3 Moule FP Further Pure 5548993  Further Pure an 3 Differentiating inverse trigonometric functions Throughout the course you have graually been increasing the number of functions
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationChapter 6: EnergyMomentum Tensors
49 Chapter 6: EnergyMomentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energymomentum tensors, then applies these ieas to the case of Bohm's moel.
More informationG4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const.
G4003 Avance Mechanics 1 The Noether theorem We alreay saw that if q is a cyclic variable, the associate conjugate momentum is conserve, q = 0 p q = const. (1) This is the simplest incarnation of Noether
More informationExperiment 2, Physics 2BL
Experiment 2, Physics 2BL Deuction of Mass Distributions. Last Upate: 20090503 Preparation Before this experiment, we recommen you review or familiarize yourself with the following: Chapters 46 in Taylor
More informationTransreal Limits and Elementary Functions
Transreal Limits an Elementary Functions Tiago S. os Reis, James A. D. W. Anerson Abstract We exten all elementary functions from the real to the transreal omain so that they are efine on ivision by zero.
More informationUsing the definition of the derivative of a function is quite tedious. f (x + h) f (x)
Derivative Rules Using te efinition of te erivative of a function is quite teious. Let s prove some sortcuts tat we can use. Recall tat te efinition of erivative is: Given any number x for wic te limit
More informationBLOWUP FORMULAS FOR ( 2)SPHERES
BLOWUP FORMULAS FOR 2)SPHERES ROGIER BRUSSEE In this note we give a universal formula for the evaluation of the Donalson polynomials on 2)spheres, i.e. smooth spheres of selfintersection 2. Note that
More informationLATTICEBASED DOPTIMUM DESIGN FOR FOURIER REGRESSION
The Annals of Statistics 1997, Vol. 25, No. 6, 2313 2327 LATTICEBASED DOPTIMUM DESIGN FOR FOURIER REGRESSION By Eva Riccomagno, 1 Rainer Schwabe 2 an Henry P. Wynn 1 University of Warwick, Technische
More informationMATH 319, WEEK 2: Initial Value Problems, Existence/Uniqueness, FirstOrder Linear DEs
MATH 319, WEEK 2: Initial Value Problems, Existence/Uniqueness, FirstOrder Linear DEs 1 InitialValue Problems We have seen that differential equations can, in general, given rise to multiple solutions.
More informationMATHEMATICS BONUS FILES for faculty and students
MATHMATI BONU FIL for faculty an stuents http://www.onu.eu/~mcaragiu1/bonus_files.html RIVD: May 15, 9 PUBLIHD: May 5, 9 toffel 1 Maxwell s quations through the Major Vector Theorems Joshua toffel Department
More informationA Course in Machine Learning
A Course in Machine Learning Hal Daumé III 12 EFFICIENT LEARNING So far, our focus has been on moels of learning an basic algorithms for those moels. We have not place much emphasis on how to learn quickly.
More informationFlash Card Construction Instructions
Flash Car Construction Instructions *** THESE CARDS ARE FOR CALCULUS HONORS, AP CALCULUS AB AND AP CALCULUS BC. AP CALCULUS BC WILL HAVE ADDITIONAL CARDS FOR THE COURSE (IN A SEPARATE FILE). The left column
More information2.5 The Chain Rule Brian E. Veitch
2.5 The Chain Rule This is our last ifferentiation rule for this course. It s also one of the most use. The best way to memorize this (along with the other rules) is just by practicing until you can o
More informationPermanent vs. Determinant
Permanent vs. Determinant Frank Ban Introuction A major problem in theoretical computer science is the Permanent vs. Determinant problem. It asks: given an n by n matrix of ineterminates A = (a i,j ) an
More information6 Wave equation in spherical polar coordinates
6 Wave equation in spherical polar coorinates We now look at solving problems involving the Laplacian in spherical polar coorinates. The angular epenence of the solutions will be escribe by spherical harmonics.
More informationIMPLICIT DIFFERENTIATION
Mathematics Revision Guies Implicit Differentiation Page 1 of Author: Mark Kulowski MK HOME TUITION Mathematics Revision Guies Level: AS / A Level AQA : C4 Eecel: C4 OCR: C4 OCR MEI: C3 IMPLICIT DIFFERENTIATION
More informationEVALUATING HIGHER DERIVATIVE TENSORS BY FORWARD PROPAGATION OF UNIVARIATE TAYLOR SERIES
MATHEMATICS OF COMPUTATION Volume 69, Number 231, Pages 1117 1130 S 00255718(00)011200 Article electronically publishe on February 17, 2000 EVALUATING HIGHER DERIVATIVE TENSORS BY FORWARD PROPAGATION
More informationELEC3114 Control Systems 1
ELEC34 Control Systems Linear Systems  Moelling  Some Issues Session 2, 2007 Introuction Linear systems may be represente in a number of ifferent ways. Figure shows the relationship between various representations.
More informationChapter 7. Integrals and Transcendental Functions
7. The Logarithm Define as an Integral Chapter 7. Integrals an Transcenental Functions 7.. The Logarithm Define as an Integral Note. In this section, we introuce the natural logarithm function using efinite
More informationComputing Derivatives J. Douglas Child, Ph.D. Rollins College Winter Park, FL
Computing Derivatives by J. Douglas Chil, Ph.D. Rollins College Winter Park, FL ii Computing Inefinite Integrals Important notice regaring book materials Texas Instruments makes no warranty, either express
More informationNOTES ON EULERBOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy,
NOTES ON EULERBOOLE SUMMATION JONATHAN M BORWEIN, NEIL J CALKIN, AND DANTE MANNA Abstract We stuy a connection between EulerMacLaurin Summation an Boole Summation suggeste in an AMM note from 196, which
More informationTractability results for weighted Banach spaces of smooth functions
Tractability results for weighte Banach spaces of smooth functions Markus Weimar Mathematisches Institut, Universität Jena ErnstAbbePlatz 2, 07740 Jena, Germany email: markus.weimar@unijena.e March
More information4.4. CHAIN RULE I 351
4.4. CHAIN RULE I 35 4.4 Chain Rule I The chain rule is perhaps the most important of the ifferentiation rules. It is immensely rich in application, an very elegantly state when notation is chosen wisely.
More informationThe continuity equation
Chapter 6 The continuity equation 61 The equation of continuity It is evient that in a certain region of space the matter entering it must be equal to the matter leaving it Let us consier an infinitesimal
More informationdx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)
Math 7 Activit: Implicit & Logarithmic Differentiation (Solutions) Implicit Differentiation. For each of the following equations, etermine x. a. tan x = x 2 + 2 tan x] = x x x2 + 2 ] = tan x] + tan x =
More information1 Introuction In the past few years there has been renewe interest in the nerson impurity moel. This moel was originally propose by nerson [2], for a
Theory of the nerson impurity moel: The Schrieer{Wol transformation re{examine Stefan K. Kehrein 1 an nreas Mielke 2 Institut fur Theoretische Physik, uprecht{karls{universitat, D{69120 Heielberg, F..
More informationThe Kepler Problem. 1 Features of the Ellipse: Geometry and Analysis
The Kepler Problem For the Newtonian 1/r force law, a miracle occurs all of the solutions are perioic instea of just quasiperioic. To put it another way, the twoimensional tori are further ecompose into
More informationIt's often useful to find all the points in a diagram that have the same voltage. E.g., consider a capacitor again.
177 (SJP, Phys 22, Sp ') It's often useful to fin all the points in a iagram that have the same voltage. E.g., consier a capacitor again. V is high here V is in between, here V is low here Everywhere
More informationOutline. Calculus for the Life Sciences II. Introduction. Tides Introduction. Lecture Notes Differentiation of Trigonometric Functions
Calculus for the Life Sciences II c Functions Joseph M. Mahaffy, mahaffy@math.ssu.eu Department of Mathematics an Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State
More informationTHE GENUINE OMEGAREGULAR UNITARY DUAL OF THE METAPLECTIC GROUP
THE GENUINE OMEGAREGULAR UNITARY DUAL OF THE METAPLECTIC GROUP ALESSANDRA PANTANO, ANNEGRET PAUL, AND SUSANA A. SALAMANCARIBA Abstract. We classify all genuine unitary representations of the metaplectic
More informationPure Further Mathematics 1. Revision Notes
Pure Further Mathematics Revision Notes June 20 2 FP JUNE 20 SDB Further Pure Complex Numbers... 3 Definitions an arithmetical operations... 3 Complex conjugate... 3 Properties... 3 Complex number plane,
More informationExperiment I Electric Force
Experiment I Electric Force Twentyfive hunre years ago, the Greek philosopher Thales foun that amber, the harene sap from a tree, attracte light objects when rubbe. Only twentyfour hunre years later,
More informationRobustness and Perturbations of Minimal Bases
Robustness an Perturbations of Minimal Bases Paul Van Dooren an Froilán M Dopico December 9, 2016 Abstract Polynomial minimal bases of rational vector subspaces are a classical concept that plays an important
More informationHarmonic Modelling of Thyristor Bridges using a Simplified Time Domain Method
1 Harmonic Moelling of Thyristor Briges using a Simplifie Time Domain Metho P. W. Lehn, Senior Member IEEE, an G. Ebner Abstract The paper presents time omain methos for harmonic analysis of a 6pulse
More informationThe total derivative. Chapter Lagrangian and Eulerian approaches
Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function
More information3.6. Implicit Differentiation. Implicitly Defined Functions
3.6 Implicit Differentiation 205 3.6 Implicit Differentiation 5 2 25 2 25 2 0 5 (3, ) Slope 3 FIGURE 3.36 The circle combines the graphs of two functions. The graph of 2 is the lower semicircle an passes
More informationMath Review for Physical Chemistry
Chemistry 362 Spring 27 Dr. Jean M. Stanar January 25, 27 Math Review for Physical Chemistry I. Algebra an Trigonometry A. Logarithms an Exponentials General rules for logarithms These rules, except where
More informationarxiv: v1 [physics.classph] 20 Dec 2017
arxiv:1712.07328v1 [physics.classph] 20 Dec 2017 Demystifying the constancy of the ErmakovLewis invariant for a time epenent oscillator T. Pamanabhan IUCAA, Post Bag 4, Ganeshkhin, Pune  411 007, Inia.
More informationParameter estimation: A new approach to weighting a priori information
Parameter estimation: A new approach to weighting a priori information J.L. Mea Department of Mathematics, Boise State University, Boise, ID 83725555 Email: jmea@boisestate.eu Abstract. We propose a
More informationGCD of Random Linear Combinations
JOACHIM VON ZUR GATHEN & IGOR E. SHPARLINSKI (2006). GCD of Ranom Linear Combinations. Algorithmica 46(1), 137 148. ISSN 01784617 (Print), 14320541 (Online). URL https://x.oi.org/10.1007/s0045300600721.
More informationThe Starting Point: Basic Concepts and Terminology
1 The Starting Point: Basic Concepts and Terminology Let us begin our stu of differential equations with a few basic questions questions that any beginner should ask: What are differential equations? What
More informationTOEPLITZ AND POSITIVE SEMIDEFINITE COMPLETION PROBLEM FOR CYCLE GRAPH
English NUMERICAL MATHEMATICS Vol14, No1 Series A Journal of Chinese Universities Feb 2005 TOEPLITZ AND POSITIVE SEMIDEFINITE COMPLETION PROBLEM FOR CYCLE GRAPH He Ming( Λ) Michael K Ng(Ξ ) Abstract We
More informationSYNCHRONOUS SEQUENTIAL CIRCUITS
CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents
More information7. Localization. (d 1, m 1 ) (d 2, m 2 ) d 3 D : d 3 d 1 m 2 = d 3 d 2 m 1. (ii) If (d 1, m 1 ) (d 1, m 1 ) and (d 2, m 2 ) (d 2, m 2 ) then
7. Localization To prove Theorem 6.1 it becomes necessary to be able to a enominators to rings (an to moules), even when the rings have zeroivisors. It is a tool use all the time in commutative algebra,
More information54 Electrostatic Boundary Value Problems
11/8/4 Section 54 Electrostatic Bounary Value Problems blank 1/ 54 Electrostatic Bounary Value Problems Reaing Assignment: pp. 149157 Q: A: We must solve ifferential equations, an apply bounary conitions
More informationIntegration: Using the chain rule in reverse
Mathematics Learning Centre Integration: Using the chain rule in reverse Mary Barnes c 999 University of Syney Mathematics Learning Centre, University of Syney Using the Chain Rule in Reverse Recall that
More informationEquilibrium in Queues Under Unknown Service Times and Service Value
University of Pennsylvania ScholarlyCommons Finance Papers Wharton Faculty Research 12014 Equilibrium in Queues Uner Unknown Service Times an Service Value Laurens Debo Senthil K. Veeraraghavan University
More informationLie symmetry and Mei conservation law of continuum system
Chin. Phys. B Vol. 20, No. 2 20 020 Lie symmetry an Mei conservation law of continuum system Shi ShenYang an Fu JingLi Department of Physics, Zhejiang SciTech University, Hangzhou 3008, China Receive
More information