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An inexact differential equation is a differential equation of the form (see also: inexact differential)
The solution to such equations came with the invention of the integrating factor by Leonhard Euler in 1739. [1]
In order to solve the equation, we need to transform it into an exact differential equation. In order to do that, we need to find an integrating factor to multiply the equation by. We'll start with the equation itself. , so we get . We will require to satisfy . We get
After simplifying we get
Since this is a partial differential equation, it is mostly extremely hard to solve, however in some cases we will get either or , in which case we only need to find with a first-order linear differential equation or a separable differential equation, and as such either
or
Differential equations arise in many problems in physics, engineering, and other sciences. The following examples show how to solve differential equations in a few simple cases when an exact solution exists.
The wave equation is a second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Due to the fact that the second order wave equation describes the superposition of an incoming wave and an outgoing wave it is also called "Two-way wave equation".
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well.
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
In multivariate calculus, a differential is said to be exact or perfect, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q.
In mathematics and its applications, classical Sturm–Liouville theory is the theory of real second-order linear ordinary differential equations of the form:
The Feynman–Kac formula named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. In 1947 when Kac and Feynman were both on Cornell faculty, Kac attended a presentation of Feynman's and remarked that the two of them were working on the same thing from different directions. The Feynman–Kac formula resulted, which proves rigorously the real case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still unproven.
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential. This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.
In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering.
Euler–Bernoulli beam theory is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral loads only. By ignoring the effects of shear deformation and rotatory inertia, it is thus a special case of Timoshenko beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.
In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.
In mathematics, an inseparable differential equation is an ordinary differential equation that cannot be solved by using separation of variables. To solve an inseparable differential equation one can employ a number of other methods, like the Laplace transform, substitution, etc.
In physics and fluid mechanics, a Blasius boundary layer describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow. Falkner and Skan later generalized Blasius' solution to wedge flow, i.e. flows in which the plate is not parallel to the flow.
An inexact differential or imperfect differential is a type of differential used in thermodynamics to express changes in path dependent quantities. In contrast, an integral of an exact differential is always path independent since the integral acts to invert the differential operator. Consequently, a quantity with an inexact differential cannot be expressed as a function of only the variables within the differential; i.e. its value cannot be inferred just by looking at the initial and final states of a given system. Inexact differentials are primarily used in calculations involving heat and work because they are path functions, not state functions.
A differential equation can be homogeneous in either of two respects.
In fluid dynamics, the Falkner–Skan boundary layer describes the steady two-dimensional laminar boundary layer that forms on a wedge, i.e. flows in which the plate is not parallel to the flow. It is a generalization of the Blasius boundary layer.
Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
