First-order Differential Equations
First-order differential equations are mathematical equations involving a function and its derivative. They are used to model various real-world phenomena and are fundamental in fields such as physics, engineering, and economics. Solutions to first-order differential equations can be found using various methods, including separation of variables, integrating factors, and exact equations.
8 Key excerpts on "First-order Differential Equations"
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...12 Difference and differential equations Many economic processes demonstrate dynamic behavior. However, in some of these cases, the underlying system is guided by rather simple relationships between quantities and their rates of change. Whenever we have a relationship between functions and their derivatives or rates of change, we essentially deal with differential or difference equations. For functions of continuous variables, we consider differential equations. For quantities dependent on a discrete variable, we talk about difference equations. Consider an economic process developing in time. Most likely the functions involved in the process will depend on their previous values. For instance, the current supply depends on the demand we observed a week, a month or a year ago. Whenever we have a relationship involving quantities at time t and their previous values at times t − 1, t − 2, and so on, we implicitly deal with a mathematical structure known as a difference equation. This chapter begins with the mathematical theory of such equations, considers some applications of difference equations to economic problems, and closes with differential equations. 12.1 Difference equations 12.1.1 Linear first-order difference equations A linear first-order difference equation may be written as follows: where For example, the equation is a linear first-order difference equation. Alternatively, the first-order difference equation in (12.1) may be written in the form: If and are constants, then we can write equation (12.3) as follows: where b and c are some constants. The difference equation is known as a first-order difference equation with constant coefficients...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...15 Solution of Differential Equations Any equation containing differential coefficients is termed differential equation. Such equations can be divided into two major types – ordinary and partial; said classification depends on whether they encompass algebraic operations, lumped in f, on a single independent variable – and thus only ordinary derivatives appear as coefficients, i.e. f { x, y, dy / dx, d 2 y / dx 2, … } = 0, or more than one independent variable – so partial derivatives play the role of independent coefficients, e.g. f { x, y, z, ∂z / ∂x, ∂z / ∂y, ∂ 2 z / ∂x 2, ∂ 2 z / ∂x∂y, ∂ 2 z / ∂y 2, …} = 0 in the bivariate case. When a differential equation contains no terms on its independent variable(s) only, it is said to be homogeneous – otherwise it is labeled as nonhomogeneous; solutions for the latter may often be obtained using the corresponding homogeneous solution as template. The order of a differential equation is the order of the highest differential coefficient contained therein; the degree of a differential equation is the power to which the highest order differential coefficient is raised when the equation is rationalized (i.e. following removal of fractional powers). The general solution of a typical partial differential equation is a combination of arbitrary functions of specific arguments; their exact form is to be determined by application of boundary (and/or initial) conditions – dependent on the nature of the system/process under scrutiny. Conversely, the general solution of an ordinary differential equation contains as many specific independent functions (each one multiplied by an arbitrary constant) as its order. The most common methods of analytical solution, via integration, will be reviewed below. 15.1 Ordinary Differential Equations A differential equation is said to be linear when it is linear in the dependent variable and its derivatives – otherwise it is said to be nonlinear...
- Bernard Liengme(Author)
- 2008(Publication Date)
- Academic Press(Publisher)
...Chapter 14 Differential Equations Publisher Summary Differential equations occur in many physical problems. This chapter opens up with the explanation of some of these problems. To solve dy/dx =f(x,y) over the x range [a, b], the value is needed to know of y(a), which is called the initial value. Problems of this type are called initial value problems. With second-order differential equations two integration constants arise. For an initial value problem one needs to know the initial value of the two values of the dependent variables. Alternatively, the problem may be defined by specifying some conditions at one value of x and others at another value of x. Such problems are called boundary value problems. The chapter describes Euler’s method to solve initial value problems. It also takes into account the Runge-Kutta method. Differential equations occur in many physical problems. Let us look at some simple examples. (i) A body falling through the air is subjected to two forces: gravity acting downward and air resistance acting upward. The first force is constant, but the second is proportional to the body's velocity. This gives rise to the first-order differential equation (14.1) (ii) Consider the chemical reaction A + B → C where the rate of reaction is proportional to the concentration of A and to the concentration of B. Let x be the amount of A and B reacted at time t, and let the initial concentration of A and B be a and b, respectively. These quantities will be related by (14.2) (iii) The equation of motion for a harmonic oscillator is (14.3) Equations 14.1 and 14.2 are examples of First-order Differential Equations, while Equation 14.3 is of second order. The equations in these examples may readily be solved by analytical means...
eBook - ePubHydraulic Modelling: An Introduction
Principles, Methods and Applications
- Pavel Novak, Vincent Guinot, Alan Jeffrey, Dominic E. Reeve(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...Chapter 2 Theoretical background – mathematics 2.1 Ordinary differential equations 2.1.1 Definitions Physical situations described by quantities that vary continuously with respect to their position in space and possibly with time can usually be described in terms of systems of partial differential equations (PDEs). These are equations that relate the quantities involved to some of their derivatives with respect to space variables and time. In the simplest case, when only a single quantity u (t) depending on a variable t is involved, the variation of u (t) with respect to t is described by an ordinary differential equation (ODE) that relates u (t) to some of its derivatives. If the highest order derivative involved in an ODE is d n u /d t n, the ODE is said to be of the nth order. The variable t is called the independent variable, and in physical situations t is often the time, while the quantity u (t) is called the dependent variable because its value depends on t. A general nth-order ODE can be written symbolically as F (t, u, u ′, u ″, … u (n)) = 0, (2.1) where u ′ = d u d t, u ″ = d 2 u d t 2 =, …, u (n) = d n u d t n, and F is an arbitrary function of its n + 1 arguments t, u, u′, u″, …,u (n). The form of equation (2.1) is too general to be of use when discussing ODEs, so in practice it is necessary to restrict study to some of the most frequently occurring types of ODE that arise in applications. This involves considering special forms that may be taken by the function F, although only the most important of these will be mentioned here. The simplest type of ODE is of the form d y / d t = g (y) h (t), where g (y) and h (t) are functions of their respective arguments. An ODE of this type is said to have separable variables, because it can be written as ∫ (1 g (y)) d y = ∫ h (t) d t, in which the variables y and t have been separated, after which the general solution follows by integration...
eBook - ePubMATLAB® Essentials
A First Course for Engineers and Scientists
- William Bober(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...10 Numerical Integration of Ordinary Differential Equations 10.1 Introduction Many ordinary differential equations (ODE) result from a particular physical law. The physical law is a mathematical model of some particular physical phenomenon. Many of the equations that have been used in this book are based on Newton’s second law of motion. For example, the equations used to describe the motion of a free falling ball in a gravitational field (Example 2.7) or the motion of the mass in a mass-spring-dashpot system (Exercise E2.3-for a complete derivation of the governing equations, see Project P2.5 in Reference 1), or the velocity and position of the basketball (Exercise E2.4) are differential equations based on Newton’s second law. The voltage in a parallel RLC circuit (Exercise E2.6) resulted from several electrical laws, including Kirchhoff’s current law, which resulted in an ordinary differential equation whose solution is given in Equation 2.13 (for a complete derivation of the governing equations, see P2.7 in Reference 1). Ordinary differential equations can be broken up into two categories: 1. Initial value problems are those in which the initial conditions of the variables are known. All of the examples and exercises mentioned above fall into this category. Additional examples include launching a rocket with a known initial position and velocity or the value of a circuit node voltage (or its slope) at t = 0. In this chapter, we only cover the initial value problem along with MATLAB ® ’s built-in ode45 function to solve these types of problems. 2. Boundary value problems in which we know variable conditions at specific coordinates in the problem geometry. For example, determining the temperature at various positions along a bar when the end temperatures at the bar ends are known...
eBook - ePub- Richard C. Dorf, Ronald J. Tallarida(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...10 Differential Equations 1. First Order-First Degree Equations M (x, y) d x + N (x, y) d y = 0 a. If the equation can be put in the form A (x) dx + B (y) dy = 0, it is separable and the solution follows by integration: ∫ A (x) d x + ∫ B (y) d y = C ; thus, x (1 + y 2) dx + ydy = 0 is separable since it is equivalent to xdx + ydy /(1 + y 2)=0, and integration. yields x 2 / 2 + 1 2 log (1 + y 2) + C = 0. b. If M (x, y) and N (x, y) are homogeneous and of the same degree in x and y, then substitution of υ x for y (thus, dy = υ dx + xd υ) will yield a separable equation in the variables x and y. [A function such as M (x, y) is homogeneous of degree n in x and y if M (cx, cy) = c n M (x, y).]For example, (y −2 x) dx + (2 y + x) dy has M and N each homogeneous and of degree one so that substitution of y = υx yields the separable equation 2 x d x + 2 υ + 1 υ 2 + υ − 1 d υ = 0. c. If M (x, y) dx + N (x, y) dy is the differential of some. function F (x, y), then the given equation is said to be exact. A necessary and sufficient condition for exactness is ∂ M / ∂ y = ∂ N / ∂ x. When the equation is exact, F is found from the relations ∂F / ∂x = M and ∂F / ∂y = N, and the solution is F (x, y = C (constant). For example, (x 2 + y) dy + (2 xy − 3 x 2) dx is exact since ∂M / ∂y = 2 x and ∂N / ∂x = 2 x. F is found from ∂F / ∂x = 2 xy − 3 x 3 and ∂F / ∂y = x 2 + y. From the first of these, F = x 2 y − x 3 + ϕ (y); from the second, F = x 2 y + y 2 /2 + ¥(x). It follows that F = x 2 y = x 3 + y 2 /2, and F = C is the solution. d. Linear, order one in y : Such an equation has the form dy + P (x) ydx = Q (x) dx. Multiplication by exp[∫ P (x) dx ] yields d [ y exp (∫ P d x) ] = Q (x) exp (∫ P d x) d x. For example, dy + (2/ x) ydy = x 2 dx is linear in y. P (x) = 2/ x, so ∫ Pdx = 2 ln x = ln x 2, and exp(∫ Pdx) = x 2...
eBook - ePub- Adil H. Mouhammed(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...I t and C t in the income equation, we obtain Y t = 20 + 0.9Y t−1 + 100 + 12 + 0.09Y t−1 − 0.09Y t−2. Y t = 132 + 0.99Y t−1 − 0.09Y t−2. This equation is a second-order linear nonhomogeneous difference equation because it takes the form Y t − 0.99Y t−1 + 0.09Y t−2 = 132, whose quadratic equation is b 2 − 0.99b + 0.09 = 0. Students are encouraged to find the time path function of Y t, that is, the solution of the above equation. (For the particular solution, try Y t = k). First-Order Linear Differential Equations The first order linear differential equation takes the form dy/dt + u(t)y = w(t), where u(t) and w(t) are both functions oft. If w(t) is equal to zero, the equation is said to be homogeneous ; otherwise, it is nonhomogeneous. Also, these two functions may be constant. If they are constant, the differential equation is said to be of constant coefficient and constant term, and this equation shall be used in this section. The differential equation is said to be of first order, because there is the first derivative dy/dt only. It is said to be of first degree, because the derivative dy/dt is raised to the first power; if the derivative is raised to the second power, (dy/dt) 2, the equation is termed second degree. If there was a second derivative d 2 y/dt 2 in lieu of the first derivative, the equation would be called of second order. In this section we are interested in finding a solution to this differential equation. Similar to the first-order difference equation, we try a solution to the homogeneous part of the equation and try a particular solution to the nonhomogeneous part. Then the two solutions are combined to give the complete solution. Example 1: Solve dy/dt + ay t = 0 (or dy/dt = − ay t), given the initial condition y o = 5. Solution: Try the solution y t = Ae −at, where A can be obtained from the initial condition. Then dy/dt = − aAe -at...
eBook - ePub- Adil H. Mouhammed(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...be PV = ∫ o 5 1000 e − 0.02 t = (− 1000 / 0.02) e − 0.02 t = − 1000 / 0.02 [ e − 0.02 (5) − e − 0.02 (0) ] = 4759. The definite integration can also be used in probability theory. For instance, if f(x) is a probability function, one can find the probability that the random variable (x) is between 1 and 3. That is, P (1 < x < 3) = ∫ f (x) dx. So, if f(x) = 0.2, then P (1 < x < 3) = ∫ 1 3 0.2 dx = 0.2 x ] = 0.2 (3) − 0.2 (1) = 0.4. First-Order Linear Difference Equations A difference equation is used to find the time path of a variable in discrete time, that is, over one day, one month, one year, and so forth. This dynamic analysis is also called period analysis. If a variable y is taken into consideration, the first difference between two periods is written as Δ y = y t - y t - 1, where y t means the value of the variable y in time t, and y t-1 means the value of the variable y in time t-1; if t is Monday, then t-1 is Sunday. If Δy is equal to 4, then the pattern of change can be described as Δ y = y t - y t - 1 = 4, which is a first-order nonhomogeneous linear difference equation. If Δy is written as Δ y = y t - y t - 1 = 0, the equation is called a linear first-order homogeneous difference equation. At any rate, the general form of a nonhomogeneous linear difference equation is y t = a 1 y t - 1 + a 2 y t - 2 + a 3 y t - 3 +... + a n y t - n + c. In this section we are interested in finding a solution for a linear first order difference equation (Chiang 1984; Goldberg 1971; Gondolfo 1971). The starting point is the first-order linear homogeneous difference equation taking the form y t - a y t − 1 = 0, where (a) is a constant coefficient. This equation is linear, because the variables y t and y t-1 are raised to the first power. The equation is of first order, because the difference between the variables y t and y t-1 is one period...
