Technology & Engineering
Exact Differential Equation
An exact differential equation is a type of differential equation where the total differential of a function can be expressed as a combination of the differentials of its variables. This means that the equation can be solved by finding a function whose total differential matches the given equation. Exact differential equations are important in engineering and physics for modeling various physical phenomena.
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3 Key excerpts on "Exact Differential Equation"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 2 Types of Differential Equations 1. Differential equation Visualization of heat transfer in a pump casing, by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution. A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines. ________________________ WORLD TECHNOLOGIES ________________________ Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly. An example of modelling a real world problem using differential equations is determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is constant but air resistance may be modelled as proportional to the ball's velocity. This means the ball's acceleration, which is the derivative of its velocity, depends on the velocity.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Types of Differential Equations 1. Differential equation Visualization of heat transfer in a pump casing, by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution. A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines. ________________________ WORLD TECHNOLOGIES ________________________ Differential equations arise in many areas of science and technology, specifically when-ever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly. An example of modelling a real world problem using differential equations is deter-mination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is constant but air resistance may be modelled as proportional to the ball's velocity. This means the ball's acceleration, which is the derivative of its velocity, depends on the velocity.
eBook - ePubDifferential Equations
Theory,Technique and Practice with Boundary Value Problems
- Steven G. Krantz(Author)
- 2015(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 1What Is a Differential Equation?
- The concept of a differential equation
- Characteristics of a solution
- Finding a solution
- Separable equations
- First-order linear equations
- Exact equations
- Orthogonal trajectories
1.1 Introductory Remarks
A differential equation is an equation relating some function f to one or more of its derivatives. An example is(1.1.1)d 2fdx 2( x )+ 2 xd fd x( x )+f 2( x ) = sin x .Observe that this particular equation involves a function f together with its first and second derivatives. Any given differential equation may or may not involve f or any particular derivative of f . But, for an equation to be a differential equation, at least some derivative of f must appear. The objective in solving an equation like (3.1.1) is to find the function f . Thus we already perceive a fundamental new paradigm: When we solve an algebraic equation, we seek a number or perhaps a collection of numbers; but when we solve a differential equation we seek one or more functions .As a simple example, consider the differential equationy ′= y .It is easy to determine that any function of the form y =Cexis a solution of this equation, for the derivative of the function is equal to itself. So we see that the solution set of this particular differential equation is an infinite family of functions parametrized by a parameter C . This phenomenon is quite typical of what we will see when we solve differential equations in the chapters that follow.Many of the laws of nature—in physics, in engineering, in chemistry, in biology, and in astronomy—find their most natural expression in the language of differential equations. Put in other words, differential equations are the language of nature. Applications of differential equations also abound in mathematics itself, especially in geometry and harmonic analysis and modeling. Differential equations occur in economics and systems science and other fields of mathematical science.
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