Table of Contents
Table of Contents
“Exercises of Ordinary Differential Equations”
INTRODUCTION
THEORETICAL OUTLINE
EXERCISES
“Exercises of Ordinary Differential Equations”
“Exercises of Ordinary Differential Equations”
SIMONE MALACRIDA
In this book, exercises are carried out regarding the following mathematical topics:
solving differential equations of various orders
systems of differential equations
Cauchy and Neumann initial value problems.
Initial theoretical hints are also presented to make the conduct of the exercises understandable.
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – THEORETICAL OUTLINE
Introduction and definitions
Resolution methods
Solutions
Remarkable differential equations
Autonomous systems
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II – EXERCISES
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
Exercise 23
Exercise 24
Exercise 25
Exercise 26
Exercise 27
Exercise 28
Exercise 29
Exercise 30
Exercise 31
Exercise 32
Exercise 33
Exercise 34
Exercise 35
Exercise 36
Exercise 37
INTRODUCTION
INTRODUCTION
In this workbook, some examples of calculations relating to ordinary differential equations are carried out.
Furthermore, the main theorems used in differential analysis of equations are presented.
Ordinary differential equations represent a fundamental point in mathematical analysis as, through their resolution, it is possible to answer many physical and technological problems.
In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.
What is presented in this workbook is generally addressed in advanced mathematical analysis courses (analysis 2).
I
THEORETICAL OUTLINE
THEORETICAL OUTLINE
Introduction and definitions
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A differential equation is a relationship between a function and some of its derivatives.
An equation defined in an interval of the set of real numbers in which the total derivatives of the function with respect to the unknown are present is called ordinary.
The order of the highest derivative in the equation is called the order of the equation.
One can generalize the defining set of an ordinary differential equation into an open and connected generic contained in the complex space of dimension greater than two.
A solution or integral of the ordinary differential equation is a function that satisfies the equation's relation.
An equation is said to be autonomous if the relation does not explicitly depend on the variable and is said to be written in normal form if it can be made explicit with respect to the derivative of maximum degree.
The equation is called linear if the solution is a linear combination of the derivatives according to this formula:
The term r(x) is called source and, if it is zero, the linear differential equation is called homogeneous .
In general, an ordinary differential equation of degree n has n linearly independent solutions, and any linear combination of them is itself a solution.
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Given a general solution of the homogeneous equation associated with an ordinary differential equation it is possible to find a particular solution of the equation.
This will be clarified shortly by the analytical methods of solving differential equations.
An ordinary differential equation of order n expressed in normal form can be reduced to a system of ordinary differential equations of order one in normal form, through the so-called first order reduction procedure.
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Resolution methods
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The ordinary differential equation of order n can be expressed as follows:
We can define coefficients such that:
Then we obtain the following equivalent system for solving the starting equation:
To study systems of ordinary equations of order n it is necessary to define the Wronskian ie the determinant of the square matrix constructed by placing the functions in the first row, the first derivative of each function in the second row and so on.
If the Wronskian is nonzero at any point in a given interval, then the associated functions are linearly independent and so are the solutions of the differential equation.
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Boundary conditions of a differential equation are defined as those conditions that the solution of the differential equation must satisfy on the boundary of the definition set.
The necessity of the boundary conditions is due to the fact that a differential equation can admit infinite solutions and only the boundary conditions allow to identify a particular solution which can be unique under suitable hypotheses.
There are different types of boundary conditions, but the most common are those of Dirichlet or those of Neumann.
A Dirichlet boundary condition , or of the first kind, places fixed boundary values for the value of the function.
A Neumann boundary condition , or of the second kind, places fixed boundary values for the value of the first derivative of the function.
Mixed or Robin boundary conditions are also defined , which fix values for the function in a given part of the boundary and for the first derivative of the function in the complementary part of the boundary.
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Initial value problems can be defined , i.e. an ordinary differential equation with a fixed value of the function at a point in the domain: this value is called the initial condition.
An initial value problem can be summarized as follows:
A Cauchy problem is given by the solution of a differential equation of order n which satisfies n-1 different initial conditions.
The theorem of existence and uniqueness of a Cauchy problem states that the solution of this problem exists and is unique, provided that the function respects certain hypotheses.
In particular, the function must be at least continuous in the neighborhood of each point relating to the initial conditions, be Lipschitz with respect to the variable y and uniform with respect to that x.
Under these assumptions, the Cauchy problem is equivalent to the Volterra integral equation, which we will discuss in the next chapters.
This theorem guarantees local but not global solutions.
To guarantee global solutions, for every initial condition there must exist a single maximum in the open interval of its neighborhood such that every solution satisfying the initial condition is a restriction of the global solution.
More generally, the Cauchy-Kovalevskaya theorem demonstrates that if the unknown and the initial conditions of a Cauchy problem are locally analytic functions then an analytic solution exists and is unique.
The analytically computable solutions of the differential equations are called exact . In many cases it is not possible to obtain exact solutions, but it is necessary to resort to numerical methods of resolution.
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The method of variations of constants allows you to determine the general integral of a linear differential equation of any order.
This method consists in substituting the variable in the original differential equation, recalling the Leibnitz rule for the product of the derivatives.
For example, for second order linear equations it is possible to make this substitution with respect to the two original solutions:
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 23.04.2023
ISBN: 978-3-7554-4005-5
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