Articles

8.6: Higher Order Derivatives


8.6: Higher Order Derivatives

The Finite Element Method: Its Basis and Fundamentals

The Finite Element Method: Its Basis and Fundamentals offers a complete introduction to the basis of the finite element method, covering fundamental theory and worked examples in the detail required for readers to apply the knowledge to their own engineering problems and understand more advanced applications.

This edition sees a significant rearrangement of the book’s content to enable clearer development of the finite element method, with major new chapters and sections added to cover:

Focusing on the core knowledge, mathematical and analytical tools needed for successful application, The Finite Element Method: Its Basis and Fundamentals is the authoritative resource of choice for graduate level students, researchers and professional engineers involved in finite element-based engineering analysis.


6+ Hours of Video Instruction
An introduction to the calculus behind machine learning models

Overview

Calculus for Machine Learning LiveLessons introduces the mathematical field of calculus—the study of rates of change—from the ground up. It is essential because computing derivatives via differentiation is the basis of optimizing most machine learning algorithms, including those used in deep learning such as backpropagation and stochastic gradient descent. Through the measured exposition of theory paired with interactive examples, you’ll develop a working understanding of how calculus is used to compute limits and differentiate functions. You’ll also learn how to apply automatic differentiation within the popular TensorFlow 2 and PyTorch machine learning libraries. Later lessons build on single-variable derivative calculus to detail gradients of learning (which are facilitated by partial-derivative calculus) and integral calculus (which determines the area under a curve and comes in handy for myriad tasks associated with machine learning).

Skill Level
Intermediate

Learn How To
--Develop an understanding of what’s going on beneath the hood of machine learning algorithms, including those used for deep learning.
--Compute the derivatives of functions, including by using AutoDiff in the popular TensorFlow 2 and PyTorch libraries.
--Be able to grasp the details of the partial-derivative, multivariate calculus that is common in machine learning papers and in many other subjects that underlie ML, including information theory and optimization algorithms.
--Use integral calculus to determine the area under any given curve, a recurring task in ML applied, for example, to evaluate model performance by calculating the ROC AUC metric.

Who Should Take This Course
--People who use high-level software libraries (e.g., scikit-learn, Keras, TensorFlow) to train or deploy machine learning algorithms and would like to understand the fundamentals underlying the abstractions, enabling them to expand their capabilities
--Software developers who would like to develop a firm foundation for the deployment of machine learning algorithms into production systems
--Data scientists who would like to reinforce their understanding of the subjects at the core of their professional discipline
--Data analysts or AI enthusiasts who would like to become data scientists or data/ML engineers, and so are keen to deeply understand the field they’re entering from the ground up (a very wise choice!)

Course Requirements
--Mathematics: Familiarity with secondary school–level mathematics will make the class easier to follow along with. If you are comfortable dealing with quantitative information, such as understanding charts and rearranging simple equations, you should be well prepared to follow along with all the mathematics.
--Programming: All code demos are in Python, so experience with it or another object-oriented programming language would be helpful for following along with the hands-on examples.

Lesson Descriptions:
Lesson 1, “Orientation to Calculus”: In Lesson 1, Jon defines calculus by distinguishing between differential and integral calculus. This is followed by a brief history of calculus that runs all the way through the modern applications, with a particular emphasis on its application to machine learning.

Lesson 2, “Limits”: Lesson 2 begins with a discussion of continuous versus discontinuous functions. Then Jon covers evaluating limits by both factoring and approaching methods. Next, he discusses what happens to limits when approaching infinity. The lesson concludes with comprehension exercises.

Lesson 3, “Differentiation”: In Lesson 3 Jon focuses on differential calculus. He covers the delta method for finding the slope of a curve and using it to derive the most common representation of a differentiation. After Jon takes a quick look at derivative notation, he introduces the most common differentiation rules: the constant rule, the power rule, the constant product rule, and the sum rule. Exercises wind up the lesson.

Lesson 4, “Advanced Differentiation Rules”: Lesson 4 continues differentiation, covering its advanced rules. These include the product rule, the quotient rule, and the chain rule. After some exercises Jon unleashes the might of the power rule in situations where you have a series of functions chained together.

Lesson 5, “Automatic Differentiation”: Lesson 5 enables you to move beyond differentiation by hand to scaling it up through automatic differentiation. This is accomplished through the PyTorch and TensorFlow libraries. After representing a line as a graph you will apply automatic differentiation to fitting that line to data points with machine learning.

Lesson 6, “Partial Derivatives”: Lesson 6 delves into partial derivatives. Jon begins with simple derivatives of multivariate functions, followed by more advanced geometrical examples, partial derivative notation, and the partial derivative chain rule.

Lesson 7, “Gradients”: Lesson 7 covers the gradient, which captures the partial derivative of cost with respect to all the parameters of the machine learning model from the previous lessons. To understand this, Jon performs a regression on individual data points and the partial derivatives of the quadratic cost. From there, he discusses what it means to descend the gradient of cost and describes the derivation of the partial derivatives of mean squared error, which enables you to learn from batches of data instead of individual points.

Lesson 8, “Integrals”: Lesson 8 switches to integral calculus. To set up a machine learning problem that requires integration to solve it, Jon starts off with binary classification problems, the confusion matrix, and ROC curve. With that problem in mind, Jon then covers the rules of indefinite and definite integral calculus needed to solve it. Next, Jon shows you how to do integration computationally. You learn how to use Python to find the area under the ROC curve. Finally, he ends the lessons with some resources for further study.


8.6: Higher Order Derivatives

In this post we will see Problems in Higher Mathematics by V. P. Minorsky.

The list of topics covered is quite exhaustive and the book has over 2500 problems and solutions. The topics covered are plane and solid analytic geometry, vector algebra, analysis, derivatives, integrals, series, differential equations etc. A good reference for those looking for many problems to solve.

The book was translated from the Russian by Yuri Ermolyev and was first published by Mir Publishers in 1975.

PDF | OCR | Cover | 600 dpi | Bookmarked | Paginated | 16.4 MB (15.6 MB Zipped) | 408 pages

(Note: IA file parameters maybe different.)

You can get the book here (IA) and here (filecloud).

Password, if needed: mirtitles

See FAQs for password related problems.

Chapter I.
Plane Analytic Geometry 11

1.1. Coordinates of a Point on· a Straight Line and in a Plane. The Distance Between Two Points 11
1.2. Dividing a Line Segment in a Given Ratio. The Area of a Triangle and a Polygon 13
1.3. The Equation of a Line as a Locus of Points 15
1.4. The Equation of a Straight Line: (1) Slope-Intercept Form, (2) General Form, (3) Intercept Form 17
1.5. The Angle Between Two Straight Lines. The Equation of a Pencil of Straight Lines Passing Through a Given Point. The Equation of a Straight Line Passing Through Two Given Points. The Point of Intersection of Two Straight Lines 20
1.6. The Normal Equation of a Straight Line. The Distance of a Point from a Straight Line. Equations of Bisectors. The Equations of a Pencil of Straight Lines Passing Through the Point of Intersection of Two Given Straight Lines 24
1.7. Miscellaneous Problems 26
1.8. The Circle 28
1.9. The Ellipse 30
1.10. The Hyperbola 33
1.11. The Parabola 37
1.12. Directrices, Diameters, and Tangents to Curves of the Second Order 41
1.13. Transformation of Cartesian Coordinates 44
1.14. Miscellaneous Problems on Second-Order Curves 49
1.15. General Equation of a Second-Order Curve 51
1.16. Polar Coordinates 57
1.17. Algebraic Curves of the Third and Higher Orders 61
1.18. Transcendental Curves 63

Chapter 2.
Vector Algebra 64

2.1. Addition of Vectors. Multiplication of a Vector by a Scalar 64
2.2. Rectangular Coordinates of a Point and a Vector in Space 68
2.3. Scalar Product of Two Vectors 71
2.4. Vector Product of Two Vectors 75
2.5. Scalar Triple Product 78

Chapter 3.
Solid Analytic Geometry 81

3.1. The Equation of a Plane 81
3.2. Basic Problems Involving the Equation of a Plane. 83
3.3. Equations of a Straight Line in Space 86
3.4. A Straight Line and a Plane 89
3.5. Spherical and Cylindrical Surfaces 92
3.6. Conical Surfaces and Surfaces of Revolution 95
3.7. The Ellipsoid, Hyperboloids, and Paraboloids 97

Chapter 4.
Higher Algebra 101

4.1. Determinants 101
4.2. Systems of First-Degree Equations 104
4.3. Complex Numbers 108
4.4. Higher-Degree Equations. Approximate Solution of Equations 111

Chapter 5.
Introduction to Mathematical Analysis 116

5.1. Variable Quantities and Functions 116
5.2. Number Sequences. Infinitesimals and Infinities. The Limit of a Variable. The Limit of a Function 120
5.3. Basic Properties of Limits. Evaluating the Indeterminate Forms 0/0 infty/ infty 126
5.4. The Limit of the Ratio sin(x)/x as x–> infty a 128
5.5. Indeterminate Expressions of the Form infty –> infty 129
5.6. Miscellaneous Problems on Limits 129
5.7. Comparison of Infinitesimals 130
5.8. The Continuity of a Function 132
5.9. Asymptotes 136
5.10. The Number e 137

Chapter 6.
The Derivative and the Differential 139

6.1. The Derivatives of Algebraic and Trigonometric Functions 139
6.2. The Derivative of a Composite Function 141
6.3. The Tangent Line and the Normal to a Plane Curve 142
6.4. Cases of Non-differentiability of a Continuous Function 145
6.5. The Derivatives of Logarithmic and Exponential Functions 147
6.6. The Derivatives of Inverse Trigonometric Functions 149
6.7. The Derivatives of Hyperbolic Functions 150
6.8. Miscellaneous Problems on Differentiation 151
6.9. Higher-Order Derivatives 151
6.10. The Derivative of an Implicit Function 154
6.11. The Differential of a Function 156
6.12. Parametric Equations of a Curve 158

Chapter 7.
Applications of the Derivative 161

7.1. Velocity and Acceleration 161
7.2. Mean-Value Theorems 163
7.3. Evaluating Indeterminate Forms. L’Hospital’s Rule 166
7.4. Increase and Decrease of a Function. Maxima and Minima 168
7.5. Finding Greatest and Least Values of a Function 172
7.6. Direction of Convexity and Points of Inflection of a Curve. Construction of Graphs 174

Chapter 8.
The Indefinite Integral 177

8.1. Indefinite Integral. Integration by Expansion 177
8.2. Integration by Substitution and Direct Integration 179
8.3. Integrals of the form dx and Those Reduced to Them 181
8.4. Integration by Parts 183
8.5. Integration of Some Trigonometric Functions 184
8.6. Integration of Rational Algebraic Functions 186
8.7. Integration of Certain Irrational Algebraic Functions 188
8.8. Integration of Certain Transcendental Functions 190
8.9. Integration of Hyperbolic Functions. Hyperbolic Substitutions 192
8.10. Miscellaneous Problems on Integration 193

Chapter 9.
The Definite Integral 195

9.1. Computing the Definite Integral 195
9.2. Computing Areas 199
9.3. The Volume of a Solid of Revolution 201
9.4. The Arc Length of a Plane Curve 203
9.5. The Area of a Surface of Revolution 205
9.6. Problems in Physics 206
9.7. ImproperIntegrals 209
9.8. The Mean Value of a Function 212
9.9. Trapezoid Rule and Simpson’s Formula 213

Chapter 10.
Curvature of Plane and Space Curves 216

10.1. Curvature of a Plane Curve. The Centre and Radius of Curvature. The Evolute of a Plane Curve 216
10.2.The Arc Length of a Space Curve 218
10.3. The Derivative of a Vector Function of a Scalar Argument and Its Mechanical and Geometrical Interpretations. The Natural Trihedron of a Curve 218
10.4. Curvature and Torsion of a Space Curve 222

Chapter 11.
Partial Derivatives, Total Differentials, and Their Applications 224

11.1. Functions of Two Variables and Their Geometrical Representation 224
11.2. Partial Derivatives of the First Order 227
11.3. Total Differential of the First Order22 8
11.4. The Derivative of a Composite Function 230
11.5. Derivatives of Implicit Functions 232
11.6. Higher-Order Partial Derivatives and Total Differentials 234
11.7. Integration of Total Differentials 237
11.8. Singular Points of a Plane Curve 239
11.9. The Envelope of a Family of Plane Curves 240
11.10. The Tangent Plane and the Normal to a Surface 241
11.11. Scalar Field. Level Lines and Level Surfaces. A Derivative Along a Given Direction. Gradient 243
11.12. The Extremum of a Function of Two Variables 245

Chapter 12.
Differential Equations 248

12.1. Fundamentals 248
12.2. First-Order Differential Equation with Variables Separable. Orthogonal Trajectories 250
12.3. First-Order Differential Equations: (I) Homogeneous, (2) Linear, (3) Bernoulli’s 253
12.4. Differential Equations Containing Differentials of a Product or a Quotient 255
12.5. First-Order Differential Equations in Total Differentials. Integrating Factor 255
12.6. First-Order Differential Equations Not Solved for the Derivative. Lagrange’s and Clairaut’s Equations 257
12.7. Differential Equations of Higher Orders Allowing for Reduction of the Order 259
12.8. Linear Homogeneous Differential Equations with Cons- tant Coefficients 261
12.9. Linear Non-homogeneous Differential Equations with Constant Coefficients 262
12.10. Differential Equations of Various Types 265
12.11. Euler’s Linear Differential Equation 266
12.12. Systems of Linear Differential Equations with Constant Coefficients 266
12.13. Partial Differential Equations of the Second Order (the Method of Characteristics) 267

Chapter 13.
Double, Triple, and Line Integrals 269

13.1. Computing Areas by Means of Double Integrals 269
13.2. The Centre of Gravity and the Moment of Inertia of an Area with Uniformly Distributed Mass (for Density mu = 1) 271
13.3. Computing Volumes by Means of Double Integrals 273
13.4. Areas of Curved Surfaces 274
13.5. The Triple Integral and Its Applications 275
13.6. The Line Integral. Green’s Formula 277
13.7. Surface Integrals. Ostrogradsky’s and Stokes’ Formulas 281

14.1. Numerical Series 285
14.2. Uniform Convergence of a Functional Series 288
14.3. Power Series 290
14.4. Taylor’s and Maclaurin’s Series 292
14.5. The Use of Series for Approximate Calculations 295
14.6. Taylor’s Series for a Function of Two Variables 298
14.7. Fourier Series. Fourier Integral 299


Contents

Suppose that f is a function of more than one variable. For instance,

The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the x z -plane, and those that are parallel to the y z -plane (which result from holding either y or x constant, respectively).

Basic definition Edit

The function f can be reinterpreted as a family of functions of one variable indexed by the other variables:

In other words, every value of y defines a function, denoted fy , which is a function of one variable x. [a] That is,

In this section the subscript notation fy denotes a function contingent on a fixed value of y, and not a partial derivative.

Once a value of y is chosen, say a, then f(x,y) determines a function fa which traces a curve x 2 + ax + a 2 on the x z -plane:

In this expression, a is a constant, not a variable, so fa is a function of only one real variable, that being x. Consequently, the definition of the derivative for a function of one variable applies:

The above procedure can be performed for any choice of a. Assembling the derivatives together into a function gives a function which describes the variation of f in the x direction:

This is the partial derivative of f with respect to x. Here is a rounded d called the partial derivative symbol. To distinguish it from the letter d, is sometimes pronounced "partial".

In general, the partial derivative of an n-ary function f(x1, …, xn) in the direction xi at the point (a1, …, an) is defined to be:

∂ f ∂ x i ( a 1 , … , a n ) = lim h → 0 f ( a 1 , … , a i + h , … , a n ) − f ( a 1 , … , a i , … , a n ) h >>(a_<1>,ldots ,a_)=lim _,ldots ,a_+h,ldots ,a_)-f(a_<1>,ldots ,a_,dots ,a_)>>> .

In the above difference quotient, all the variables except xi are held fixed. That choice of fixed values determines a function of one variable

In other words, the different choices of a index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.

This vector is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f which takes the point a to the vector ∇f(a). Consequently, the gradient produces a vector field.

Formal definition Edit

∂ ∂ x i f ( a ) = lim h → 0 f ( a 1 , … , a i − 1 , a i + h , a i + 1 , … , a n ) − f ( a 1 , … , a i , … , a n ) h = lim h → 0 f ( a + h e i ) − f ( a ) h >>f(mathbf )&=lim _,ldots ,a_,a_+h,a_,ldots ,a_)-f(a_<1>,ldots ,a_,dots ,a_)>>&=lim _ )-f(mathbf )>>end>>

Even if all partial derivatives ∂f/∂xi(a) exist at a given point a, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that f is a C 1 function. This can be used to generalize for vector valued functions, f : U → R m ^> , by carefully using a componentwise argument.


Calculus 1 — Single variable calculus

Chapter Index
Disk 1

Section 1: What Is A Derivative?
Section 2: The Derivative Defined As A Limit
Section 3: Differentiation Formulas
Section 4: Derivatives Of Trigonometric Functions
Section 5: The Chain Rule
Section 6: Higher Order Derivatives
Section 7: Related Rates
Section 8: Curve Sketching Using Derivatives

Disk 2
Section 9: Introduction To Integrals
Section 10: Solving Integrals
Section 11: Integration By Substitution
Section 12: Calculating Volume With Integrals
Section 13: Derivatives and Integrals Of Exponentials
Section 14: Derivatives Of Logarithms
Section 15: Integration By Parts
Section 16: Integration By Trig Substitution
Section 17: Improper Integrals

Calculus 2 — Advanced Calculus

Chapter Index
Disk 1
Section 1: Inverse Trigonometric Functions
Section 2: Derivatives of Inverse Trigonometric Functions
Section 3: Hyperbolic Functions
Section 4: Inverse Hyperbolic Functions
Section 5: L'Hospital's Rule
Section 6: Trigonometric Integrals

Disk 2
Section 7: Integration By Partial Fractions
Section 8: Arc Length
Section 9: Area Of A Surface Of Revolution
Section 10: Parametric Equations
Section 11: Arc Length In Parametric Equations
Section 12: Surface Area Of Revolution In Parametric Equations

Disk 3
Section 13: Polar Coordinates
Section 14: Polar Equations
Section 15: Area And Length In Polar Coordinates
Section 16: Sequences


Disk 4
Section 17: Series
Section 18: Integral Test Of Series Convergence
Section 19: Comparison Tests Of Series Convergence
Section 20: Alternating Series Test Of Convergence
Section 21: Ratio and Root Test Of Series Convergence

Calculus 3 Vol 1 — Multivariate Calculus

Disk 1
Section 1: 3D Cartesian Coordinates
Section 2: Introduction To Vectors
Section 3: The Vector Dot Product
Section 4: The Vector Cross Product
Section 5: Vector Valued Functions

Disk 2
Section 6: Multivariable Functions And Partial Derivatives
Section 7: The Chain Rule For Partial Derivatives
Section 8: The Directional Derivative

Disk 3
Section 9: The Gradient
Section 10: Double Integrals
Section 11: Double Integrals In Polar Coordinates

Calculus 3 Vol 2 — Multivariate Calculus

Disk 1
Section 1: Triple Integrals
Section 2: Triple Integrals In Cylindrical Coordinates

Disk 2
Section 3: Triple Integrals In Spherical Coordinates
Section 4: Divergence And Curl Of A Vector Field
Section 5: Line Integrals

Disk 3
Section 6: Line Integrals In A Vector Field
Section 7: Alternative Form Of Line Integrals In Vector Fields
Section 8: Fundamental Theorem Of Line Integrals

Disk 4
Section 9: Green's Theorem
Section 10: Surface Integrals
Section 11: Flux Integrals
Section 12: Stokes Theorem
Section 13: The Divergence Theorem


Generalization of the Multipoint meshless FDM application to the nonlinear analysis

The paper focuses on the new Multipoint meshless finite difference method, following the original Collatz higher order multipoint concept and the essential ideas of the Meshless FDM. The method was formulated, developed, and tested for various boundary value problems. Generalization of the multipoint method application to nonlinear analysis is the purpose of this research.

The first attempt of the multipoint technique application to the geometrically nonlinear problems was successfully done recently. The case of physically nonlinear problem is considered in this paper. Several benefits of the proposed approach are highlighted, numerical algorithm and selected results are presented, and application of the multipoint method to nonlinear analysis is summarized.


Dense Trajectories and Motion Boundary Descriptors for Action Recognition

This paper introduces a video representation based on dense trajectories and motion boundary descriptors. Trajectories capture the local motion information of the video. A dense representation guarantees a good coverage of foreground motion as well as of the surrounding context. A state-of-the-art optical flow algorithm enables a robust and efficient extraction of dense trajectories. As descriptors we extract features aligned with the trajectories to characterize shape (point coordinates), appearance (histograms of oriented gradients) and motion (histograms of optical flow). Additionally, we introduce a descriptor based on motion boundary histograms (MBH) which rely on differential optical flow. The MBH descriptor shows to consistently outperform other state-of-the-art descriptors, in particular on real-world videos that contain a significant amount of camera motion. We evaluate our video representation in the context of action classification on nine datasets, namely KTH, YouTube, Hollywood2, UCF sports, IXMAS, UIUC, Olympic Sports, UCF50 and HMDB51. On all datasets our approach outperforms current state-of-the-art results.

This is a preview of subscription content, access via your institution.


Intermediate to advanced Perl programmers

  • Preface
  • 1. Recursion and Callbacks
    • 1.1 Decimal to Binary Conversion
    • 1.2 Factorial
      • 1.2.1 Why Private Variables are Important
      • 1.7.1 More Flexible Selection
      • 1.8.1 Fibonacci Numbers
      • 1.8.2 Partitioning
      • 2.1 Configuration File Handling
        • 2.1.1 Table-driven configuration
        • 2.1.2 Advantages of Dispatch Tables
        • 2.1.3 Dispatch Table Strategies
        • 2.1.4 Default Actions
        • 2.2.1 HTML Processing Revisited
        • 3.1 Caching Fixes Recursion
        • 3.2 Inline Caching
          • 3.2.1 Static Variables
          • 3.5.1 Scope and Duration
            • 3.5.1.1 Scope
            • 3.5.1.2 Duration
            • 3.6.1 Functions whose Return Values do not Depend on their Arguments
            • 3.6.2 Functions with Side Effects
            • 3.6.3 Functions that Return References
            • 3.6.4 A Memoized Clock?
            • 3.6.5 Very Fast Functions
            • 3.7.1 More Applications of User-Supplied Key Generators
            • 3.7.2 Inlined Cache Manager with Argument Normalizer
            • 3.7.3 Functions with Reference Arguments
            • 3.7.4 Partioning
            • 3.7.5 Custom Key Generation for Impure Functions
            • 3.8.1 Memoization of Object Methods
            • 3.12.1 Profiling and Performance Analysis
            • 3.12.2 Automatic Profiling
            • 3.12.3 Hooks
            • 4.1 Introduction
              • 4.1.1 Filehandles are Iterators
              • 4.1.2 Iterators are Objects
              • 4.1.3 Other Common Examples of Iterators
              • 4.2.1 A Trivial Iterator: upto()
                • 4.2.1.1 Syntactic Sugar for Manufacturing Iterators
                • 4.3.1 Permutations
                • 4.3.2 Genomic Sequence Generator
                • 4.3.3 Filehandle Iterators
                • 4.3.4 A Flat-File Database
                  • 4.3.4.1 Improved Database
                  • 4.3.5.1 A Query Package that Transforms Iterators
                  • 4.3.5.2 An Iterator that Reads Files Backwards
                  • 4.3.5.3 Putting it Together
                  • 4.4.1 imap()
                  • 4.4.2 igrep()
                  • 4.4.3 list_iterator()
                  • 4.4.4 append()
                  • 4.5.1 Avoiding the Problem
                  • 4.5.2 Alternative undefs
                  • 4.5.3 Rewriting Utilities
                  • 4.5.4 Iterators that Return Multiple Values
                  • 4.5.5 Explicit Exhaustion Function
                  • 4.5.6 Four-Operation Iterators
                  • 4.5.7 Iterator Methods
                  • 4.6.1 Using foreach to Loop over more than one Array
                  • 4.6.2 An Iterator with an each-like Interface
                  • 4.6.3 Tied Variable Interfaces
                    • 4.6.3.1 Summary of tie
                    • 4.6.3.2 Tied Scalars
                    • 4.6.3.3 Tied Filehandle
                    • 4.7.1 Pursuing only Interesting Links
                    • 4.7.2 Referring URLs
                    • 4.7.3 robots.txt
                    • 4.7.4 Summary
                    • 5.1 The Partition Problem Revisited
                      • 5.1.1 Finding All Possible Partions
                      • 5.1.2 Optimizations
                      • 5.1.3 Variations
                      • 5.4.1 Tail Call Elimination
                        • 5.4.1.1 Someone Else's Problem
                        • 5.4.1.2 Creating Tail Calls
                        • 5.4.1.3 Explicit Stacks
                          • 5.4.1.3.1 Eliminating Recursion from fib()
                          • 6.1 Linked Lists
                          • 6.2 Lazy Linked Lists
                            • 6.2.1 A Trivial Stream: upto()
                            • 6.2.2 Utilities for Streams
                            • 6.3.1 Memoizing Streams
                            • 6.5.1 Generating Strings in Order
                            • 6.5.2 Regex Matching
                            • 6.5.3 Cutsorting
                              • 6.5.3.1 Log Files
                              • 6.6.1 Approximation Streams
                              • 6.6.2 Derivatives
                              • 6.6.3 The Tortoise and the Hare
                              • 6.6.4 Finance
                              • 6.7.1 Derivatives
                              • 6.7.2 Other Functions
                              • 6.7.3 Symbolic Computation
                              • 7.1 Currying
                              • 7.2 Common Higher-Order Functions
                                • 7.2.1 Automatic Currying
                                • 7.2.2 Prototypes
                                  • 7.2.2.1 Prototype Problems
                                  • 7.3.1 Boolean Operators
                                  • 7.4.1 Operator Overloading
                                  • 8.1 Lexers
                                    • 8.1.1 Emulating the operator
                                    • 8.1.2 Lexers More Generally
                                    • 8.1.3 Chained Lexers
                                    • 8.1.4 Peeking
                                    • 8.2.1 Grammars
                                    • 8.2.2 Parsing Grammars
                                    • 8.3.1 Very Simple Parsers
                                    • 8.3.2 Parser Operators
                                    • 8.3.3 Compound Operators
                                    • 8.4.1 A Calculator
                                    • 8.4.2 Left Recursion
                                    • 8.4.3 A Variation on star()
                                    • 8.4.4 Generic Operator Parsers
                                    • 8.4.5 Debugging
                                    • 8.4.6 The Finished Calculator
                                    • 8.4.7 Error Diagnosis and Recovery
                                      • 8.4.7.1 Error Recovery Parsers
                                      • 8.4.7.2 Exceptions
                                      • 8.7.1 The Lexer
                                      • 8.7.2 The Parser
                                      • 8.8.1 Continuations
                                      • 8.8.2 Parse Streams
                                      • 9.1 Constraint Systems
                                      • 9.2 Local Propagation Networks
                                        • 9.2.1 Implementing a Local Propagation Network
                                        • 9.2.2 Problems with Local Propagation
                                        • 9.4.1 Equations
                                          • 9.4.1.1 ref($base) || $base
                                          • 9.4.1.2 Solving Equations
                                          • 9.4.1.3 Constraints
                                          • 9.4.2.1 Constant Values
                                          • 9.4.2.2 Tuple Values
                                          • 9.4.2.3 Feature Values
                                          • 9.4.2.4 Intrinsic Constraints
                                          • 9.4.2.5 Synthetic Constraints
                                          • 9.4.2.6 Feature Value Methods
                                          • 9.4.3.1 Scalar Types
                                          • 9.4.3.2 Type methods
                                          • 9.4.4.1 Parser Extensions
                                          • 9.4.4.2 %TYPES
                                          • 9.4.4.3 Programs
                                          • 9.4.4.4 Definitions
                                          • 9.4.4.5 Declarations
                                          • 9.4.4.6 Expressions
                                          • 9.4.5 Missing Features

                                          Higher Order Linear Homogeneous Differential Equations with Constant Coefficients – Page 2

                                          Thus, the equation has two roots ( = 1,) ( = 5,) the first of which has multiplicity (2.) Then the general solution of differential equations can be written as follows:

                                          Example 3.

                                          Write the characteristic equation:

                                          Factor the left side and find the roots:

                                          Note that one of the roots of the cubic polynomial is the number (lambda = -1.) Therefore, we divide ( <+ 2>) by (lambda + 1:)

                                          As a result, the characteristic equation takes the following form:

                                          We find the roots of the quadratic equation:

                                          Thus, the characteristic equation has four distinct roots, two of which are complex:

                                          The general solution of the differential equation can be represented as

                                          where (, ldots, ) are arbitrary constants.

                                          Example 4.

                                          The characteristic equation can be written as

                                          Factor the left side and calculate the roots:

                                          As it can be seen, the equation has the following roots:

                                          and imaginary roots have multiplicity (2.) In accordance with the rules set out above, we write the general solution in the form

                                          where (, ldots, ) are arbitrary numbers.

                                          Example 5.

                                          Calculate the roots of the characteristic equation

                                          We see that the roots of the equation are equal

                                          The first root is of multiplicity (2.) The general solution of the differential equation is given by