Mathematics
Nonhomogeneous Differential Equation
A nonhomogeneous differential equation is a type of differential equation that includes a function on the right-hand side that is not equal to zero. This function is typically a forcing function or input to the system being modeled. Nonhomogeneous differential equations are solved using methods such as variation of parameters, undetermined coefficients, or Laplace transforms.
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6 Key excerpts on "Nonhomogeneous Differential Equation"
- S. Graham Kelly(Author)
- 2008(Publication Date)
- CRC Press(Publisher)
419 Chapter 6 Partial differential equations 6.1 Homogeneous partial differential equations A partial differential equation is a differential equation with more than one independent variable. Although the independent variables can represent a variety of coordinates and scales, this study focuses on partial differential equations in which the independent variables are spatial coordinates and time. Consider a problem in which it is desired to determine the variation of a dependent variable, say φ , in a region of space R which is bounded by a sur-face S . Let r be a position vector from the origin of the coordinate system to a particle or point in R . Most generally, r is a function of three independent spatial coordinates. The dependent variable may also be a function of time and designated as φ ( r , t ). The surface S is described by g ( r, t ) = 0. The problem is an interior problem if g is bounded and the solution is to be obtained for position vectors defined within the interior of g , as illustrated in Figure 6.1a. The problem is an exterior problem if the solution is to be obtained for posi-tion vectors defined outside of g , as illustrated in Figure 6.1b. The general form of a linear partial differential equation is L M G r φ φ φ + + = f t ( , ) (6.1) where L is a linear operator involving derivatives with respect to spatial vari-ables, M is a linear operator involving derivatives with respect to t , and G is a linear operator involving mixed partial derivatives involving time and spatial variables. Determination of a solution of Equation 6.1 requires application of appro-priate initial conditions and boundary conditions. The number of initial conditions required depends on the order of M . If M is first-order, then it is necessary to specify φ ( r ,0). If M is second-order it, is also necessary to specify ∂φ / ∂ t( r ,0). The number and type of boundary conditions depend on the order and form of L .
eBook - PDF- Stephen A. Wirkus, Randall J. Swift(Authors)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
4.1 Nonhomogeneous Equations We will now begin discussing the nonhomogeneous equation a n ( x ) d n y dx n + a n -1 ( x ) d n -1 y dx n -1 + . . . + a 1 ( x ) dy dx + a 0 ( x ) y = F ( x ) , (4.1) originally given at the beginning of Chapter 3. Up to this point we have been focusing on the solution of the homogeneous equation a n ( x ) d n y dx n + a n -1 ( x ) d n -1 y dx n -1 + . . . + a 1 ( x ) dy dx + a 0 ( x ) y = 0 , (4.2) 241 242 Chapter 4. Techniques of Nonhomogeneous Linear Eqns and concentrating on the case in which (4.2) has constant coefficients. We will see how our work with homogeneous equations will play a role. In general it is not easy to solve this equation; sometimes we can get lucky, while other times we may only be able to guess a particular solution. Fortunately, many real world phenomena are accurately modeled with sinusoidal forcing functions and we will be able to solve these with the methods of this chapter. We begin by considering (4.1) written in operator notation: P ( D ) y = F ( x ) , where P ( D ) = a n ( x ) D n + a n -1 ( x ) D n -1 + . . . + a 1 ( x ) D + a 0 ( x ) . We define a particular solution , y p , to be any solution of (4.1). This means that P ( D ) y p = F ( x ) . Example 1 Verify that y p = x 2 -2 x is a particular solution to y 00 + y 0 = 2 x. Solution We need to substitute this into the differential equation and show that it holds. 2 |{z} y 00 p + 2 x -2 | {z } y 0 p = 2 x, which clearly holds for all x . Thus y p is a particular solution. THEOREM 4.1.1 Let y p be any particular solution of the nonhomogeneous nth order linear differential equation (4.1). Let u be any solution of the corresponding homogeneous equation (4.2); then u + y p is also a solution of the given nonhomogeneous equation. Example 2 Note that y = x is a solution of the nonhomogeneous equation d 2 y dx 2 + y = x and that y = sin x is a solution of the corresponding homogeneous equation d 2 y dx 2 + y = 0 .
- 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.
- C. Pozrikidis(Author)
- 2002(Publication Date)
- CRC Press(Publisher)
Chapter 6 Inhomogeneous, nonlinear, and time-dependent problems In previous chapters, we have discussed integral representations, integral equations, and boundary-element methods for Laplace’s equation in two and three dimensions, with occasional reference to Helmholtz’s equation and the steady-state convection – diffusion equation. Prerequisites for applying the theoretical formulation and numer-ical methods to other differential equations of the general form (6.1) where is a differential operator, are the following: The differential operator is elliptic , that is, the solution of (6.1) is deter-mined exclusively by data specified around the boundaries of the solution do-main. The differential equation is homogeneous , that is, if the boundary data are zero, then the solution is also zero. The differential equation is linear , that is, if two functions satisfy the dif-ferential equation, then any linear combination of them will also satisfy the differential equation. A Green’s function of the differential equation is available in analytical or readily computable form. In real life, we encounter problems involving inhomogeneous, nonlinear, and time-dependent (parabolic or hyperbolic) equations expressing evolution from an initial state or wave propagation. Examples include the convection – diffusion equation in the presence of a distributed source, the unsteady heat conduction equation, Burgers’ equation, the Navier-Stokes equation, and the wave equation. To take advantage of the benefits of the boundary-integral formulation, we must extend the theoretical foundation and numerical implementation so that we can tackle this broader class of equations. In this chapter, we discuss the generalization of the boundary-integral formulation with emphasis on practical implementation. In Sections 6.1-6.3 we discuss the so-lution of linear inhomogeneous equations and the computation of domain integrals 131
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
In fact, later in this section we use this method to derive a formula for a particular solution of an arbitrary second-order linear Nonhomogeneous Differential Equation. On the other hand, the method of variation of parameters eventually requires us to evaluate certain integrals involving the nonhomogeneous term in the differential equation, and this may present difficulties. Before looking at this method in the general case, we illustrate its use in an example. EXAMPLE 3.6.1 Find the general solution of ′′ + 4 = 8 tan() − ∕2 < < ∕2. (1) Solution Observe that this problem is not a good candidate for the method of undetermined coefficients, as described in Section 3.5, because the nonhomogeneous term () = 8 tan() involves a quotient (rather than a sum or a product) of sin() and cos(). Therefore, the method of undetermined coefficients cannot be applied; we need a different approach. Observe also that the homogeneous equation corresponding to equation (1) is ′′ + 4 = 0, (2) and that the general solution of equation (2) is () = 1 cos(2) + 2 sin(2). (3) ▼ ▼ 146 CHAPTER 3 Second-Order Linear Differential Equations ▼ ▼ The basic idea in the method of variation of parameters is similar to the method of reduction of order introduced at the end of Section 3.4. In the general solution (3), replace the constants 1 and 2 by functions 1 () and 2 (), respectively, and then determine these functions so that the resulting expression = 1 () cos(2) + 2 () sin(2) (4) is a solution of the nonhomogeneous equation (1). To determine 1 and 2 , we need to substitute for from equation (4) in differential equa- tion (1). However, even without carrying out this substitution, we can anticipate that the result will be a single equation involving some combination of 1 , 2 , and their first two derivatives.
- N.N. Puri(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
Chapter 3 Ordinary Di ff erential and Di ff erence Equations 3.1 Introduction Linear di ff erential and di ff erence equations with constant coe ffi cients play a very important part in engineering problems. The solution of these equations is reasonably simple and most system textbooks treat the subject in a gingerly fashion. In reality, the thought process involved in the solution of these equations is of fundamental importance. The parallelism between di ff erential and di ff erence equations is emphasized. Matrix notation is introduced for its conciseness. The treatment of matrix di ff erential (or di ff erence) equations is presented here in greater detail. Furthermore, the stability of di ff erential and di ff erence equations has been studied via second method of Liapunov including an extensive table of various di ff erential equations and conditions under which the systems representing these equations are stable. 212 Ordinary Di ff erential and Di ff erence Equations 3.2 System of Di ff erential and Di ff erence Equations 3.2.1 First Order Di ff erential Equation Systems Ideas developed here are later applied to higher order systems. Consider ˙ x + ax = f ( t ) , x ( t ) | t = 0 = x 0 , a constant (3.1) where f ( t ) is a known continuous function of time (forcing function) and x is a system response, sometimes denoted as x ( t ). Equation 3.1 is called linear nonhomogeneous because the left-hand of the equation is a function of independent variable t and all the terms are linear in the dependent variable x . Method of Solution First consider the homogeneous equation ˙ x + ax = 0 (3.2) We seek a solution of the form x = x ( t ) = e λ t k (3.3) ˙ x = λ e λ t k = λ x where k and λ are unknown constants. From Eqs. 3.2 and 3.3, ( λ + a ) x = 0 (3.4) For a nontrivial solution, P ( λ ) = ( λ + a ) = 0 ⇒ λ = -a (3.5)
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