## Intermediate Course in Differential Equations

An equation containing only first derivatives is a first-order differential equation , an equation containing the second derivative is a second-order differential equation , and so on.

## Differential equations introduction

Two broad classifications of both ordinary and partial differential equations consists of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones. In the next group of examples, the unknown function u depends on two variables x and t or x and y. Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest. For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists.

• Modeling situations with differential equations?
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The solution may not be unique. See Ordinary differential equation for other results. However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:. The theory of differential equations is closely related to the theory of difference equations , in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

The study of differential equations is a wide field in pure and applied mathematics , physics , and engineering.

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All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons.

Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i. Instead, solutions can be approximated using numerical methods. Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics , differential equations are used to model the behavior of complex systems.

The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations.

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Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation , the wave equation , which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier , is governed by another second-order partial differential equation, the heat equation.

It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black—Scholes equation in finance is, for instance, related to the heat equation. The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic.

### Course Features

Main articles: Ordinary differential equation and Linear differential equation. Main article: Partial differential equation.

Maths-2B-Differential Equations Basics class by Chandra Shekhar

Main article: Non-linear differential equations. See also: Time scale calculus. Studies in the History of Mathematics and Physical Sciences. Bibcode : AmJPh.. The name field is required. Please enter your name. The E-mail message field is required. Please enter the message.

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