Linear First-Order Equations

Size: px
Start display at page:

Download "Linear First-Order Equations"

Transcription

1 5 Linear First-Orer Equations Linear first-orer 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 first-orer 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 first-orer 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 first-orer equation to be linear ) By the way, the criteria given here for a ifferential equation being linear will be extene later to higher-orer 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 higher-orer 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 first-orer ifferential equations 51 Basic Notions Definitions A first-orer 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 first-orer 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 First-Orer 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 first-orer 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 first-orer 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 first-orer 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 First-Orer 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 First-Orer Linear Equations As we just erive, the real trick to solving a first-orer 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 first-orer linear ifferential equations, + p(x)y = f (x)

5 Solving First-Orer 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 First-Orer 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 First-Orer 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 initial-value 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 First-Orer 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 First-Orer 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 initial-value 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 initial-value 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 initial-value problem (see problem 55) Thus, the above efinite integral formula gives us the one an only solution to the above initial-value 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 well-efine, 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 First-Orer 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 initial-value problems involving first-orer 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 finite-jump iscontinuities in an interval (α,β) Also let an y 0 be any two numbers with α < < β Then the initial-value 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 first-orer 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 initial-value 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 initial-value 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 finite-jump 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

The Exact Form and General Integrating Factors

The Exact Form and General Integrating Factors 7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily

More information

First Order Linear Differential Equations

First Order Linear Differential Equations LECTURE 6 First Orer Linear Differential Equations A linear first orer orinary ifferential equation is a ifferential equation of the form ( a(xy + b(xy = c(x. Here y represents the unknown function, y

More information

Implicit Differentiation

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 information

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x) Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)

More information

1 Lecture 20: Implicit differentiation

1 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 information

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1 Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of

More information

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent

More information

23 Implicit differentiation

23 Implicit differentiation 23 Implicit ifferentiation 23.1 Statement The equation y = x 2 + 3x + 1 expresses a relationship between the quantities x an y. If a value of x is given, then a corresponing value of y is etermine. For

More information

2 ODEs Integrating Factors and Homogeneous Equations

2 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 information

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+

More information

Ordinary Differential Equations

Ordinary Differential Equations Orinary Differential Equations Example: Harmonic Oscillator For a perfect Hooke s-law 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 information

Integration by Parts

Integration by Parts Integration by Parts 6-3-207 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 information

Math 1B, lecture 8: Integration by parts

Math 1B, lecture 8: Integration by parts Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores

More information

Math 1271 Solutions for Fall 2005 Final Exam

Math 1271 Solutions for Fall 2005 Final Exam Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly

More information

Higher-Order Equations: Extending First-Order Concepts

Higher-Order Equations: Extending First-Order Concepts 11 Higher-Order Equations: Extending First-Order Concepts Let us switch our attention from first-order differential equations to differential equations of order two or higher. Our main interest will be

More information

Applications of the Wronskian to ordinary linear differential equations

Applications 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 information

Mathcad Lecture #5 In-class Worksheet Plotting and Calculus

Mathcad Lecture #5 In-class Worksheet Plotting and Calculus Mathca Lecture #5 In-class Worksheet Plotting an Calculus At the en of this lecture, you shoul be able to: graph expressions, functions, an matrices of ata format graphs with titles, legens, log scales,

More information

Math 342 Partial Differential Equations «Viktor Grigoryan

Math 342 Partial Differential Equations «Viktor Grigoryan Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite

More information

Differentiation Rules Derivatives of Polynomials and Exponential Functions

Differentiation Rules Derivatives of Polynomials and Exponential Functions Derivatives of Polynomials an Exponential Functions Differentiation Rules Derivatives of Polynomials an Exponential Functions Let s start with the simplest of all functions, the constant function f(x)

More information

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises

More information

Schrödinger s equation.

Schrödinger 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 information

Euler equations for multiple integrals

Euler 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 information

Solutions to Math 41 Second Exam November 4, 2010

Solutions to Math 41 Second Exam November 4, 2010 Solutions to Math 41 Secon Exam November 4, 2010 1. (13 points) Differentiate, using the metho of your choice. (a) p(t) = ln(sec t + tan t) + log 2 (2 + t) (4 points) Using the rule for the erivative of

More information

MA 2232 Lecture 08 - Review of Log and Exponential Functions and Exponential Growth

MA 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 information

Differentiation ( , 9.5)

Differentiation ( , 9.5) Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the

More information

Exam 2 Review Solutions

Exam 2 Review Solutions Exam Review Solutions 1. True or False, an explain: (a) There exists a function f with continuous secon partial erivatives such that f x (x, y) = x + y f y = x y False. If the function has continuous secon

More information

Math 210 Midterm #1 Review

Math 210 Midterm #1 Review Math 20 Miterm # Review This ocument is intene to be a rough outline of what you are expecte to have learne an retaine from this course to be prepare for the first miterm. : Functions Definition: A function

More information

Table of Common Derivatives By David Abraham

Table of Common Derivatives By David Abraham Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec

More information

Solutions to Practice Problems Tuesday, October 28, 2008

Solutions to Practice Problems Tuesday, October 28, 2008 Solutions to Practice Problems Tuesay, October 28, 2008 1. The graph of the function f is shown below. Figure 1: The graph of f(x) What is x 1 + f(x)? What is x 1 f(x)? An oes x 1 f(x) exist? If so, what

More information

STUDENT S COMPANIONS IN BASIC MATH: THE FOURTH. Trigonometric Functions

STUDENT S COMPANIONS IN BASIC MATH: THE FOURTH. Trigonometric Functions STUDENT S COMPANIONS IN BASIC MATH: THE FOURTH Trigonometric Functions Let me quote a few sentences at the beginning of the preface to a book by Davi Kammler entitle A First Course in Fourier Analysis

More information

Implicit Differentiation. Lecture 16.

Implicit Differentiation. Lecture 16. Implicit Differentiation. Lecture 16. We are use to working only with functions that are efine explicitly. That is, ones like f(x) = 5x 3 + 7x x 2 + 1 or s(t) = e t5 3, in which the function is escribe

More information

UNDERSTANDING INTEGRATION

UNDERSTANDING INTEGRATION UNDERSTANDING INTEGRATION Dear Reaer The concept of Integration, mathematically speaking, is the "Inverse" of the concept of result, the integration of, woul give us back the function f(). This, in a way,

More information

A Brief Review of Elementary Ordinary Differential Equations

A Brief Review of Elementary Ordinary Differential Equations A A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on

More information

Fall 2016: Calculus I Final

Fall 2016: Calculus I Final Answer the questions in the spaces provie on the question sheets. If you run out of room for an answer, continue on the back of the page. NO calculators or other electronic evices, books or notes are allowe

More information

Review of Differentiation and Integration for Ordinary Differential Equations

Review of Differentiation and Integration for Ordinary Differential Equations Schreyer Fall 208 Review of Differentiation an Integration for Orinary Differential Equations In this course you will be expecte to be able to ifferentiate an integrate quickly an accurately. Many stuents

More information

4.2 First Differentiation Rules; Leibniz Notation

4.2 First Differentiation Rules; Leibniz Notation .. FIRST DIFFERENTIATION RULES; LEIBNIZ NOTATION 307. First Differentiation Rules; Leibniz Notation In this section we erive rules which let us quickly compute the erivative function f (x) for any polynomial

More information

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) = 2 2t 1 + t 2 cos x = 1 t2 We will revisit the double angle identities:

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) = 2 2t 1 + t 2 cos x = 1 t2 We will revisit the double angle identities: 6.4 Integration using tanx/) We will revisit the ouble angle ientities: sin x = sinx/) cosx/) = tanx/) sec x/) = tanx/) + tan x/) cos x = cos x/) sin x/) tan x = = tan x/) sec x/) tanx/) tan x/). = tan

More information

Final Exam Study Guide and Practice Problems Solutions

Final Exam Study Guide and Practice Problems Solutions Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making

More information

Chapter 2. Exponential and Log functions. Contents

Chapter 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 information

Vectors in two dimensions

Vectors in two dimensions Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication

More information

Integration Review. May 11, 2013

Integration Review. May 11, 2013 Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In

More information

A. Incorrect! The letter t does not appear in the expression of the given integral

A. Incorrect! The letter t does not appear in the expression of the given integral AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question

More information

Further Differentiation and Applications

Further Differentiation and Applications Avance Higher Notes (Unit ) Prerequisites: Inverse function property; prouct, quotient an chain rules; inflexion points. Maths Applications: Concavity; ifferentiability. Real-Worl Applications: Particle

More information

Section 7.1: Integration by Parts

Section 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 information

Two formulas for the Euler ϕ-function

Two formulas for the Euler ϕ-function Two formulas for the Euler ϕ-function Robert Frieman A multiplication formula for ϕ(n) The first formula we want to prove is the following: Theorem 1. If n 1 an n 2 are relatively prime positive integers,

More information

Derivatives and the Product Rule

Derivatives 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 information

Survey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013

Survey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013 Survey Sampling Kosuke Imai Department of Politics, Princeton University February 19, 2013 Survey sampling is one of the most commonly use ata collection methos for social scientists. We begin by escribing

More information

Define each term or concept.

Define 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 information

Lecture 2 Lagrangian formulation of classical mechanics Mechanics

Lecture 2 Lagrangian formulation of classical mechanics Mechanics Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,

More information

Calculus of Variations

Calculus of Variations 16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t

More information

Mathematics 116 HWK 25a Solutions 8.6 p610

Mathematics 116 HWK 25a Solutions 8.6 p610 Mathematics 6 HWK 5a Solutions 8.6 p6 Problem, 8.6, p6 Fin a power series representation for the function f() = etermine the interval of convergence. an Solution. Begin with the geometric series = + +

More information

QF101: 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, 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 information

Chapter 6: Integration: partial fractions and improper integrals

Chapter 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 information

First Order Linear Differential Equations

First 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 information

Optimization Notes. Note: Any material in red you will need to have memorized verbatim (more or less) for tests, quizzes, and the final exam.

Optimization Notes. Note: Any material in red you will need to have memorized verbatim (more or less) for tests, quizzes, and the final exam. MATH 2250 Calculus I Date: October 5, 2017 Eric Perkerson Optimization Notes 1 Chapter 4 Note: Any material in re you will nee to have memorize verbatim (more or less) for tests, quizzes, an the final

More information

1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.

1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7. Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency

More information

Antiderivatives. Definition (Antiderivative) If F (x) = f (x) we call F an antiderivative of f. Alan H. SteinUniversity of Connecticut

Antiderivatives. Definition (Antiderivative) If F (x) = f (x) we call F an antiderivative of f. Alan H. SteinUniversity of Connecticut Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Definition (Inefinite

More information

IMPLICIT DIFFERENTIATION

IMPLICIT DIFFERENTIATION IMPLICIT DIFFERENTIATION CALCULUS 3 INU0115/515 (MATHS 2) Dr Arian Jannetta MIMA CMath FRAS Implicit Differentiation 1/ 11 Arian Jannetta Explicit an implicit functions Explicit functions An explicit function

More information

Quantum Mechanics in Three Dimensions

Quantum 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 information

Differentiability, Computing Derivatives, Trig Review. Goals:

Differentiability, Computing Derivatives, Trig Review. Goals: Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an

More information

Diagonalization of Matrices Dr. E. Jacobs

Diagonalization of Matrices Dr. E. Jacobs Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is

More information

Computing Derivatives

Computing 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 information

0.1 The Chain Rule. db dt = db

0.1 The Chain Rule. db dt = db 0. The Chain Rule A basic illustration of the chain rules comes in thinking about runners in a race. Suppose two brothers, Mark an Brian, hol an annual race to see who is the fastest. Last year Mark won

More information

February 21 Math 1190 sec. 63 Spring 2017

February 21 Math 1190 sec. 63 Spring 2017 February 21 Math 1190 sec. 63 Spring 2017 Chapter 2: Derivatives Let s recall the efinitions an erivative rules we have so far: Let s assume that y = f (x) is a function with c in it s omain. The erivative

More information

Calculus in the AP Physics C Course The Derivative

Calculus in the AP Physics C Course The Derivative Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.

More information

Kramers Relation. Douglas H. Laurence. Department of Physical Sciences, Broward College, Davie, FL 33314

Kramers Relation. Douglas H. Laurence. Department of Physical Sciences, Broward College, Davie, FL 33314 Kramers Relation Douglas H. Laurence Department of Physical Sciences, Browar College, Davie, FL 333 Introuction Kramers relation, name after the Dutch physicist Hans Kramers, is a relationship between

More information

4 Exact Equations. F x + F. dy dx = 0

4 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 information

3.2 Differentiability

3.2 Differentiability Section 3 Differentiability 09 3 Differentiability What you will learn about How f (a) Might Fail to Eist Differentiability Implies Local Linearity Numerical Derivatives on a Calculator Differentiability

More information

ay (t) + by (t) + cy(t) = 0 (2)

ay (t) + by (t) + cy(t) = 0 (2) Solving ay + by + cy = 0 Without Characteristic Equations, Complex Numbers, or Hats John Tolle Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213-3890 Some calculus courses

More information

3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes

3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we

More information

Some functions and their derivatives

Some 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 information

Section The Chain Rule and Implicit Differentiation with Application on Derivative of Logarithm Functions

Section The Chain Rule and Implicit Differentiation with Application on Derivative of Logarithm Functions Section 3.4-3.6 The Chain Rule an Implicit Differentiation with Application on Derivative of Logarithm Functions Ruipeng Shen September 3r, 5th Ruipeng Shen MATH 1ZA3 September 3r, 5th 1 / 3 The Chain

More information

Markov Chains in Continuous Time

Markov Chains in Continuous Time Chapter 23 Markov Chains in Continuous Time Previously we looke at Markov chains, where the transitions betweenstatesoccurreatspecifietime- steps. That it, we mae time (a continuous variable) avance in

More information

The Principle of Least Action

The Principle of Least Action Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of

More information

The Press-Schechter mass function

The Press-Schechter mass function The Press-Schechter 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 information

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) 2t 1 + t 2 cos x = 1 t2 sin x =

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) 2t 1 + t 2 cos x = 1 t2 sin x = 6.4 Integration using tan/ We will revisit the ouble angle ientities: sin = sin/ cos/ = tan/ sec / = tan/ + tan / cos = cos / sin / tan = = tan / sec / tan/ tan /. = tan / + tan / So writing t = tan/ we

More information

Tutorial 1 Differentiation

Tutorial 1 Differentiation Tutorial 1 Differentiation What is Calculus? Calculus 微積分 Differential calculus Differentiation 微分 y lim 0 f f The relation of very small changes of ifferent quantities f f y y Integral calculus Integration

More information

Inverse Functions. Review from Last Time: The Derivative of y = ln x. [ln. Last time we saw that

Inverse Functions. Review from Last Time: The Derivative of y = ln x. [ln. Last time we saw that Inverse Functions Review from Last Time: The Derivative of y = ln Last time we saw that THEOREM 22.0.. The natural log function is ifferentiable an More generally, the chain rule version is ln ) =. ln

More information

Year 11 Matrices Semester 2. Yuk

Year 11 Matrices Semester 2. Yuk Year 11 Matrices Semester 2 Chapter 5A input/output Yuk 1 Chapter 5B Gaussian Elimination an Systems of Linear Equations This is an extension of solving simultaneous equations. What oes a System of Linear

More information

Unit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule

Unit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin

More information

SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA. where L is some constant, usually called the Lipschitz constant. An example is

SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA. where L is some constant, usually called the Lipschitz constant. An example is SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA. Uniqueness for solutions of ifferential equations. We consier the system of ifferential equations given by x = v( x), () t with a given initial conition

More information

Equations of lines in

Equations of lines in Roberto s Notes on Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1 Equations of lines in What ou nee to know alrea: The ot prouct. The corresponence between equations an graphs.

More information

x = c of N if the limit of f (x) = L and the right-handed limit lim f ( x)

x = c of N if the limit of f (x) = L and the right-handed limit lim f ( x) Limit We say the limit of f () as approaches c equals L an write, lim L. One-Sie Limits (Left an Right-Hane Limits) Suppose a function f is efine near but not necessarily at We say that f has a left-hane

More information

Differentiability, Computing Derivatives, Trig Review

Differentiability, 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 information

Trigonometric Functions

Trigonometric 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 information

Lagrangian and Hamiltonian Mechanics

Lagrangian and Hamiltonian Mechanics Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical

More information

PDE Notes, Lecture #11

PDE Notes, Lecture #11 PDE Notes, Lecture # from Professor Jalal Shatah s Lectures Febuary 9th, 2009 Sobolev Spaces Recall that for u L loc we can efine the weak erivative Du by Du, φ := udφ φ C0 If v L loc such that Du, φ =

More information

Assignment 1. g i (x 1,..., x n ) dx i = 0. i=1

Assignment 1. g i (x 1,..., x n ) dx i = 0. i=1 Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition

More information

Lecture 6: Calculus. In Song Kim. September 7, 2011

Lecture 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 information

1 Lecture 18: The chain rule

1 Lecture 18: The chain rule 1 Lecture 18: The chain rule 1.1 Outline Comparing the graphs of sin(x) an sin(2x). The chain rule. The erivative of a x. Some examples. 1.2 Comparing the graphs of sin(x) an sin(2x) We graph f(x) = sin(x)

More information

MATH 13200/58: Trigonometry

MATH 13200/58: Trigonometry MATH 00/58: Trigonometry Minh-Tam Trinh For the trigonometry unit, we will cover the equivalent of 0.7,.4,.4 in Purcell Rigon Varberg.. Right Triangles Trigonometry is the stuy of triangles in the plane

More information

Separation of Variables

Separation 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 information

f(x) f(a) Limit definition of the at a point in slope notation.

f(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. 126-235 Definition. Limit Definition of Derivatives at a point

More information

The Natural Logarithm

The Natural Logarithm The Natural Logarithm -28-208 In earlier courses, you may have seen logarithms efine in terms of raising bases to powers. For eample, log 2 8 = 3 because 2 3 = 8. In those terms, the natural logarithm

More information

Math 180, Exam 2, Fall 2012 Problem 1 Solution. (a) The derivative is computed using the Chain Rule twice. 1 2 x x

Math 180, Exam 2, Fall 2012 Problem 1 Solution. (a) The derivative is computed using the Chain Rule twice. 1 2 x x . Fin erivatives of the following functions: (a) f() = tan ( 2 + ) ( ) 2 (b) f() = ln 2 + (c) f() = sin() Solution: Math 80, Eam 2, Fall 202 Problem Solution (a) The erivative is compute using the Chain

More information

θ x = f ( x,t) could be written as

θ x = f ( x,t) could be written as 9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)

More information

Section 7.2. The Calculus of Complex Functions

Section 7.2. The Calculus of Complex Functions Section 7.2 The Calculus of Complex Functions In this section we will iscuss limits, continuity, ifferentiation, Taylor series in the context of functions which take on complex values. Moreover, we will

More information

1 Definition of the derivative

1 Definition of the derivative Math 20A - Calculus by Jon Rogawski Chapter 3 - Differentiation Prepare by Jason Gais Definition of the erivative Remark.. Recall our iscussion of tangent lines from way back. We now rephrase this in terms

More information

f(x + h) f(x) f (x) = lim

f(x + h) f(x) f (x) = lim Introuction 4.3 Some Very Basic Differentiation Formulas If a ifferentiable function f is quite simple, ten it is possible to fin f by using te efinition of erivative irectly: f () 0 f( + ) f() However,

More information

and from it produce the action integral whose variation we set to zero:

and 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 information