Mathematics
Solutions to Differential Equations
Solutions to differential equations refer to the functions that satisfy a given differential equation. These solutions can be found through various methods, such as separation of variables, integrating factors, and using specific techniques for different types of differential equations (e.g., linear, nonlinear, first-order, second-order). The solutions provide a way to model and understand the behavior of systems in various fields, including physics, engineering, and biology.
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9 Key excerpts on "Solutions to Differential Equations"
eBook - ePubMathematical Methods for Finance
Tools for Asset and Risk Management
- Sergio M. Focardi, Frank J. Fabozzi, Turan G. Bali(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
The solutions to a differential equation or system of differential equations can be as simple as explicit formulas. When an explicit formula is not possible to obtain, various numerical methods can be used to approximate a solution. Even in the absence of an exact solution, properties of solutions of a differential equation can be determined. A large number of properties of differential equations have been established over the last three centuries. In this chapter, we provide only a brief introduction to the concept of differential equations and their properties, limiting our discussion to the principal concepts. We do not cover stochastic differential equations.DIFFERENTIAL EQUATIONS DEFINED A differential equation is a condition expressed as a functional link between one or more functions and their derivatives. It is expressed as an equation (that is, as an equality between two terms). A solution of a differential equation is a function that satisfies the given condition. For example, the conditionequates to zero a linear relationship between an unknown function Y (x ), its first and second derivatives Y′ (x ), Y ″ (x ), and a known function b (x ). (In some equations, we denote the first and second derivatives by a single and double prime, respectively.) The unknown function Y (x ) is the solution of the equation that is to be determined.There are two broad types of differential equations: ordinary differential equations and partial differential equations. Ordinary differential equations are equations or systems of equations involving only one independent variable. Another way of saying this is that ordinary differential equations involve only total derivatives. In contrast, partial differential equations are differential equations or systems of equations involving partial derivatives. That is, there is more than one independent variable.ORDINARY DIFFERENTIAL EQUATIONS In full generality, an ordinary differential equation (ODE) can be expressed as the following relationship:where Y(m )(x ) denotes the m th derivative of an unknown function Y (x ). If the equation can be solved for the n
eBook - ePub- Clayton R. Paul(Author)
- 2011(Publication Date)
- Wiley-IEEE Press(Publisher)
to use mathematical skills to determine what the equations governing the system are telling us about how the system behaves.4.2 WHERE THEY ARISE: THE MEANING OF A SOLUTIONIn Chapter 1 we showed examples of where differential equations arise in electrical and mechanical engineering systems. Ordinary differential equations can be found routinely in all of engineering.How do we know that a particular formula for x(t) is a solution to the differential equation? The simple method is to substitute it into the differential equation, perform the required differentiations, and see if satisfies the equation [i.e., both sides of (4.1) are the same]. However, unlike algebraic equations, differential equations have an infinite number of possible solutions. So we need some additional information to pin down which of these is the actual solution for the problem we are investigating. This additional information comprises what are called the initial conditions for the specific problem being investigated. For example, consider the first-order equation in (4.1a). For the specific problem that this differential equation describes, we would need to specify an additional initial condition for the value of x (t) at some starting time, say t = 0, which we denote as x (0). So the complete specification of the problem would be to specify (a) the differential equation and (b) the initial condition as(4.1a)Once we have obtained a solution x(t) that satisfies this differential equation and the initial condition, we can say that the solution x(t) is valid for all t ≥ 0. Similarly, the second-order equation in (4.1b) requires two initial conditions: x(0) and its derivative att = 0,dx(t)/dt\t=0 ≡
eBook - PDF- Virginia W. Noonburg(Author)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
In some sense, first-order equations are thought of as being simpler than second-order equations. By the time you have worked through Chapter 2, you may not want to believe that this is true, and there are special cases where it definitely is not true; however, it is a useful way to distinguish between equations to which different methods of solution apply. In Chapter 4, we will see that solving ordinary differential equations of order greater than one can always be reduced to solving a system of first-order equations. 1.1.4 What is a Solution? Given a differential equation, exactly what do we mean by a solution? It is first important to realize that we are looking for a function, and therefore it needs to be defined on some interval of its independent variable. Before computers were available, a solution of a differential equation usually referred to an analytic solution; that is, a formula obtained by algebraic methods or other methods of mathematical analysis such as integration and differentiation, from which exact values of the unknown function could be obtained. 4 Chapter 1 Introduction to Differential Equations Definition 1.3. An analytic solution of a differential equation is a sufficiently differ-entiable function that, if substituted into the equation, together with the necessary deriva-tives, makes the equation an identity (a true statement for all values of the independent variable) over some interval of the independent variable. It is now possible, however, using sophisticated computer packages, to numeri-cally approximate solutions to a differential equation to any desired degree of accuracy, even if no formula for the solution can be found. You will be introduced to numerical methods in Chapter 2, and many of the equations in later chapters will only be solvable using numerical or graphical methods. Given an analytic solution, it is usually fairly easy to check whether or not it sat-isfies the equation.
- Yuefan Deng(Author)
- 2014(Publication Date)
- World Scientific(Publisher)
1 Chapter 1 First-Order Differential Equations 1.1 Definition of Differential Equations A differential equation (DE) is a mathematical equation that relates some functions of one or more variables with its derivatives. A DE is used to describe changing quantities and it plays a major role in quantitative studies in many disciplines such as all areas of engineering, physical sciences, life sciences, and economics. Examples Are they DEs or not? + + = 0 No Chapter 1 First-Order Differential Equations 2 + ′ + = 0 Yes Here ′ = + ′ + ′ = 0 Yes Here ′ = and ′ = ′′ = Yes Here ′ = To solve a DE is to express the solution of the unknown function (the dependent variable) in mathematical terms without the derivatives. Example + = 0 ′ = − is not a solution = − is a solution In general, there are two common ways in solving DEs, analytic and numerical. Most DEs, difficult to solve by analytical methods, must be “solved” by numerical methods although many DEs are too stiff to solve using numerical techniques. Solving DEs by numerical methods is a different subject requiring basic knowledge of computer programming and numerical analysis; this book focuses on introducing analytical methods for solving very small families of DEs that are truly solvable. Although the DEs are quite simple, the solution methods are not and the essential solution steps and terminologies involved are fully applicable to much more complicated DEs. Classification of DEs Classification of DEs is itself another subject in studying DEs. We will introduce classifications and terminologies for flowing the contents of the book but one may still need to lookup terms undefined here.
eBook - ePub- Martha L. Abell, James P. Braselton, Martha L. L. Abell(Authors)
- 2016(Publication Date)
- Academic Press(Publisher)
The advantages of using Mathematica in the study of differential equations are numerous, but perhaps some of the most useful include that of being able to produce the graphics associated with solutions of differential equations as well as generating very accurate numerical solutions of equations that can only be solved numerically. This is in the discussion of applications because many physical situations are modeled with differential equations. For example, we will see that the motion of a pendulum can be modeled by a differential equation. When we solve the problem of the motion of a pendulum, we use technology to actually watch the pendulum move. The same is true for the motion of a mass attached to the end of a spring as well as many other problems. In having this ability, the study of differential equations becomes much more meaningful as well as interesting.If you are a beginning Mathematica user and, especially, new to Version 10, the Appendix contains an introduction to Mathematica, including discussions about entering and evaluating commands, and taking advantage of Mathematica’s extensive help facilities.Although Chapter 1 is short in length, Chapter 1 introduces examples that will be investigated in subsequent chapters. Also, the vocabulary introduced in Chapter 1 will be used throughout the text. Consequently, even though, to a large extent, it may be read quickly, subsequent chapters will take advantage of the terminology and techniques discussed here.1.1 Definitions and Concepts
We begin our study of differential equations by explaining what a differential equation is.Definition 1 Differential EquationA differential equation is an equation that contains the derivative or differentials of one or more dependent variables with respect to one or more independent variables. If the equation contains only ordinary derivatives (of one or more dependent variables) with respect to a single independent variable, the equation is called an ordinary differential equation .Example 1.1.1 Discuss properties of the following differential equations.1.d y / d x =and dy /dx + du /dx = u + x 2 yx 2/y 2cos y2.( y − 1 ) d x + x cos y d y = 13.xy ″+ xy ′+y = 0x 2−n 24.= ( a − b y ) xd xd t= ( − m + n x ) y .d yd tSolution
eBook - ePubSymbolic Mathematics for Chemists
A Guide for Maxima Users
- Fred Senese(Author)
- 2018(Publication Date)
- Wiley(Publisher)
11 Differential Equations True laws can only be expressed in differential equations. – Bertrand Russell [77] A differential equation relates an independent variable with derivatives of a dependent variable. It has the general form: (11.1) where f explicitly depends on at least one of the derivatives of y. Such equations model many (if not most) physical processes, and solving them is a central task in reaction kinetics and quantum theory. Consider an elementary reaction. The rate of the first-order reaction is related to the concentration of A at time t by (11.2) where A is the concentration of A as an unknown function of time t. An ordinary (not partial) derivative appears in the equation, so the equation is called an ordinary differential equation, or ODE. The solution of a differential equation is a function, not the value of a variable. In this case, solving the equation gives us A as a function of t. In Section 1.2.4 we saw that the ode2 function can find solutions to ODEs: (%i1) diff (A(t),t) = -k*A(t); (%i2) ode2 (%, A(t), t); (%o1) (%o2) The first line tells Maxima that A depends on t (see Section 6.3.1). The second line specifies the differential equation. The ode2 function gives a general solution in terms of %c, which is some unknown constant. The general solution represents an entire family of solutions with different values of %c (Figure 11.5). In Section 11.1, we’ll see how to find specific values for the constant by imposing initial conditions or boundary conditions on the general solution. Equation (11.2) directly gives d A /d t, which is the slope of the tangent lines for the solutions A (t). This lets us visualize the equation geometrically as a vector field, which is easily plotted in Maxima. In Section 11.3, we’ll see how solutions are represented as trajectories through the vector field. The highest derivative that appears in the equation determines the order of the differential equation
eBook - PDFDifferential Equations
A first course on ODE and a brief introduction to PDE
- Shair Ahmad, Antonio Ambrosetti(Authors)
- 2019(Publication Date)
- De Gruyter(Publisher)
3 Analytical study of first order differential equations 3.1 General first order differential equations Before discussing methods of solving differential equations in general, in this chapter we study and analyze the theoretical aspects of such equations. In contrast to Chap-ter 2, where we mainly learned methods of solving differential equations, here we stress a more rigorous treatment of the rationale for drawing conclusions concerning the qualitative behavior of solutions. This is particularly important since there are no known methods to solve such equations in general. Of course, these general equations include the linear equations covered in Chapter 2. Let A ⊆ ℝ 2 be a set, and let f = f ( t , x ) be a real valued function defined on A . Definition 3.1. A first order ordinary differential equation is an equation of the form x ? = f ( t , x ), where x ? = dx dt . (3.1) We note that the linear equation x ? + p ( t ) x = q ( t ) , discussed in the previous chap-ter, is included in (3.1), with f ( t , x ) = − p ( t ) x + q ( t ) and A = I × ℝ , I being the interval where p and q are defined. Definition 3.2. A solution (or integral) of (3.1) is a differentiable real valued function x ( t ) defined on an interval I ⊆ ℝ such that (i) ( t , x ( t )) ∈ A for all t ∈ I ; (ii) x ? ( t ) = f ( t , x ( t )) for all t ∈ I . More precisely, to see that a differentiable function x ( t ) solves x ? = f ( t , x ) one evaluates x ? ( t ) and checks to see if x ? = f ( t , x ) holds true for all t in I . Let x ( t ), t ∈ I , be a solution to (3.1). Let us recall that the tangent line to x ( t ) at a point t ∗ ∈ I has equation x = x ? ( t ∗ )( t − t ∗ ) + x ( t ∗ ) . Since x ? ( t ∗ ) = f ( t ∗ , x ( t ∗ )) we find x = f ( t ∗ , x ( t ∗ ))( t − t ∗ ) + x ( t ∗ ). So, from a geometrical point of view, a solution to (3.1) is a differentiable curve x = x ( t ) such that the slope of the tangent line at each point t ∗ ∈ I equals f ( t ∗ , x ( t ∗ )) .
eBook - ePub- Wim van Drongelen(Author)
- 2018(Publication Date)
- Academic Press(Publisher)
Chapter 9 Differential Equations Introduction Abstract In this chapter we review ordinary differential equations (ODEs) as a tool to model dynamics. We present examples of how to formulate them based on the dynamical system that needs to be modeled, and demonstrate the mathematical techniques one can employ to solve the equation analytically. We show how to solve linear differential equations with and without a forcing term, the so-called inhomogeneous and homogeneous ODEs, respectively. To illustrate the analysis of these equations, ODEs with first-order derivatives (e.g., d c / d t) and second-order derivatives (e.g., d 2 c / d t 2) are used in the examples. Next, we show how higher-order ODEs can be represented as a set of first-order ones, and how this leads to a formalism in matrix/vector notation that can be efficiently analyzed using techniques from linear algebra. To complete the overview of the available tools for solving ODEs, the final part of this chapter briefly refers to application of Laplace and Fourier transforms (see also Chapter 12) to solve them. Keywords Characteristic equation; Dynamics; Eigenvalue; Forcing term; Linear homogeneous equation; Linear inhomogeneous equation; Ordinary differential equation (ODE) 9.1. Modeling Dynamics When modeling some aspect of a neural system, we can represent static variables with algebraic expressions, e.g., the concentration c of some chemical in the brain is 10 units, c = 10. Of course, we can make these expressions a bit more complicated; for instance, the concentration could also depend on some combination of other substances a 1 and a 2 : e.g., c = 5 a 1 + a 2 + 10. It is important to realize that the variables do not change with time or space. For that reason the value of this type of equation is limited to situations where we study properties that remain constant over the range and duration of our interest
eBook - PDFPartial Differential Equations
A First Course
- Rustum Choksi(Author)
- 2022(Publication Date)
- American Mathematical Society(Publisher)
We will most often use ? to denote the unknown function, i.e., the dependent variable. So, for example, in purely spatial variables we would be dealing with ?(?, ?), ?(?, ?, ?) , or more generally ?( x ) . When time is also relevant, we will deal with ?(?, ?), ?(?, ?, ?), ?(?, ?, ?, ?) , or more generally ?( x , ?) . 1.2. • What Are Partial Differential Equations and Why Are They Ubiquitous? “ The application of calculus to modern science is largely an exercise in the formulation, solution, and interpretation of partial differential equations . ... Even at the cutting edge of modern physics, partial dif-ferential equations still provide the mathematical infrastructure. ” -Steven Strogatz a a Strogatz is an applied mathematician well known for his outstanding undergraduate book on the qualitative theory of ODEs: Nonlinear Dynamics and Chaos , Westview Press. This quote is taken from his popular science book Infinite Powers: How Calculus Reveals the Secrets of the Universe , Houghton Mifflin Harcourt Publishing Company, 2019. We begin with a precise definition of a partial differential equation. Definition of a Partial Differential Equation (PDE) Definition 1.2.1. A partial differential equation (PDE) is an equation which relates an unknown function ? and its partial derivatives together with inde-pendent variables. In general, it can be written as ?( independent variables , ?, partial derivatives of ?) = 0, for some function ? which captures the structure of the PDE. For example, in two independent variables a PDE involving only first-order partial derivatives is described by ?(?, ?, ?, ? ? , ? ? ) = 0, (1.1) where ? ∶ ℝ 5 → ℝ . Laplace’s equation is a particular PDE which in two independent variables is associated with ?(? ?? , ? ?? ) = ? ?? + ? ?? = 0. Note that often there is no explicit appearance of the independent variables, and the function ? need not depend on all possible derivatives.
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