Nonhomogeneous Differential Equation

3 Key excerpts on "Nonhomogeneous Differential Equation"

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  eBook - ePub

  Signal Processing for Neuroscientists

  • 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

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  eBook - ePub

  Elementary Differential Equations

  Applications, Models, and Computing

  • Charles Roberts(Author)
  • 2018(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  nonhomogeneous equation. For example, y p (x) = 2 is a particular solution of the nonhomogeneous linear differential equation y ″ + 4 y = 8, since y p ″ + 4 y p = 0 + 4 (2) = 8 and y p contains no arbitrary constant. Example 9 Verification of a Particular Solution Show that y (x) = x 2 - 2 x is a particular solution of the nonhomogeneous linear differential. equation (26) y (3) - 3 y (2) + 3 y (1) - y = - x 2 + 8 x - 12. Solution Differentiating y (x) = x 2 - 2 x three times, we find y (1) (x) = 2 x - 2, y (2) (x) = 2, and y (3) = 0. Substituting for y (x) and its derivatives in the DE (26), we see that y (3) - 3 y (2) + 3 y (1) - y = 0 - 3 (2) + 3 (2 x - 2) - (x 2 - 2 x) = - 6 + 6 x - 6 - x 2 + 2 x = - x 2 + 8 x - 12. Since y (x) satisfies the DE (26) and contains no arbitrary. constant, y (x) is a particular solution of (26). A Representation Theorem for N-th Order Nonhomogeneous Linear Differential Equations If y p (x) is any particular solution on the interval I of the nonhomogeneous linear differential. equation (24) a n (x) y (n) (x) + a n - 1 (x) y (n - 1) (x) + ⋯ + a 1 (x) y (1) (x) + a 0 (x) y (x) = b (x), and. if y 1 (x), y 2 (x), …, y n (x) are n linearly independent solutions on I of the associated homogeneous. equation (25) a n (x) y (n) (x) + a n - 1 (x) y (n - 1) (x) + ⋯ + a 1 (x) y (1) (x) + a 0 (x) y (x) = 0, then every solution of the DE (24) on the interval I has the. form y (x) = c 1 y 1 (x) + c 2 y 2 (x) + ⋯ + c n y n (x) + y p (x) where c 1, c 2, …, c n are suitably chosen. constants. Proof: Let y c (x) = c 1 y 1 (x) + c 2 y 2 (x) + ⋯ + c n y n (x) where the c i are arbitrary constants and let z (x) be any solution of the nonhomogeneous linear DE (24) on I. In order to prove this theorem, we must show that it is possible to choose the c i so that z (x) = y c (x) + y p (x). Since z (x) and y p (x) are both solutions on the interval I of the nonhomogeneous DE (24), w (x) = z (x) - y p (x) is a solution on I of the associated homogeneous DE (25)

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  eBook - ePub

  Mathematical Physics

  An Introduction

  • Derek Raine(Author)
  • 2018(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  6

  DIFFERENTIAL EQUATIONS 1

  Differential equations are fundamental to physical science. The behavior of any system continuously evolving in time (the motion of a body subject to a force, a chemical reaction, ...) is governed by one or more differential equations. So is the behavior of continuous systems in space. Solving the equations means finding out how the body moves given the forces acting, how the reaction proceeds given the reagents etc. Only the simplest of equations can be solved exactly. But these are important in developing a physical intuition for the sort of behavior to expect. (Conversely a good physical insight will provide a basis for the mathematical solution.) They are also used later in the derivation of approximations to the solutions of more complex equations. You need to develop a good understanding of the equations in this chapter.

  6.1.WHAT ARE DIFFERENTIAL EQUATIONS?

  A differential equation is an equation for an unknown function, y (t ) say, of an independent variable (in this case t ) that involves one or more derivatives of y with respect to t .

  Example 6.1 Here are some examples of differential equations, from different fields.

  A simple model for population growth (with λ > 0), or radioactive decay (with λ < 0)

  Simple harmonic motion, ω a constant

  Governor of steam turbine Motion of a planet in general relativity

  Economics growth model, A, B constants

  Geophysics, f (x ) a given function of x

  The order of a differential equation is the number of the highest derivative appearing in it.

  Example 6.2 What are the orders of equations (6.1) and (6.3)?

  Equation (6.1) is first order because it contains only first derivatives (i.e. dy/dt ).

  Equation (6.3) is third order because the number of the highest derivative is 3 (i.e. d 3 y/dt 3 ).

  Exercise 6.1 What are the orders of the differential equations(6.2) and (6.4) to (6.6)?

  It might be helpful to clarify the notation that is used for differential equations. If a function y depends on time t we often write ẏ instead of dy/dt, ÿ for d 2 y/dt 2 etc. If the independent variable is not time but position, x , we may use “prime” symbols like y′ , y″ , . . . y n for dy/dx, d 2 y/dx 2 , . . . , d n y/dx n

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