Differential Equation By B.d. Sharma Pdf Book -
Melvin Amey
In this paper, we built the KHDE with future event horizon as IR cutoff in a non-flat universe with model parameters K and c. KHDE is constructed by applying Kaniadakis entropy to a cosmological framework instead of normal Bekenstein Hawking entropy and the holographic principle. We first offered a straightforward differential equation for the holographic dark energy density parameter \(\Omega _d\) in order to examine the cosmic applications of KHDE. Additionally, as a function of \(\Omega _d\), we deduced an analytical expression for the holographic dark energy equation-of-state parameter \(w_d\) while accounting for both closed and open spatial geometry. Although the aforementioned differential equation can be analytically solved in an implicit form when \(K=0\) and \(\gamma =0\), in general case it can not be solved analytically, hence one must numerically elaborate it.
In this class, an Ordinary differential equation will be discussed. In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on. The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions. Learn how to solve differential equations here.
One of the easiest ways to solve the differential equation is by using explicit formulas. In this article, let us discuss the definition, types, methods to solve the differential equation, order and degree of the differential equation, ordinary differential equations with real-word examples and a solved problem.
A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable)
A differential equation contains derivatives which are either partial derivatives or ordinary derivatives. The derivative represents a rate of change, and the differential equation describes a relationship between the quantity that is continuously varying with respect to the change in another quantity. There are a lot of differential equations formulas to find the solution of the derivatives.
You can see in the first example, it is a first-order differential equation which has degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as:
A function that satisfies the given differential equation is called its solution. The solution that contains as many arbitrary constants as the order of the differential equation is called a general solution. The solution free from arbitrary constants is called a particular solution. There exist two methods to find the solution of the differential equation.
Separation of the variable is done when the differential equation can be written in the form of dy/dx = f(y)g(x) where f is the function of y only and g is the function of x only. Taking an initial condition, rewrite this problem as 1/f(y)dy= g(x)dx and then integrate on both sides.
Differential equations have several applications in different fields such as applied mathematics, science, and engineering. Apart from the technical applications, they are also used in solving many real life problems. Let us see some differential equation applications in real-time.
The various other applications in engineering are: heat conduction analysis, in physics it can be used to understand the motion of waves. The ordinary differential equation can be utilized as an application in the engineering field for finding the relationship between various parts of the bridge.
1. An ordinary differential equation contains one independent variable and its derivatives. It is frequently called ODE. The general definition of the ordinary differential equation is of the form: Given an F, a function os x and y and derivative of y, we have
The order of the highest order derivative present in the differential equation is called the order of the equation. If the order of the differential equation is 1, then it is called the first order. If the order of the equation is 2, then it is called a second-order, and so on.
The main purpose of the differential equation is to compute the function over its entire domain. It is used to describe the exponential growth or decay over time. It has the ability to predict the world around us. It is widely used in various fields such as Physics, Chemistry, Biology, Economics and so on.
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