In mathematics, an equation that contains partial derivatives, expressing a process of change that depends on more than one independent variable. It can be read as a statement about how a process evolves without specifying the formula defining the process. Given the initial state of the process (such as its size at time zero) and a description of how it is changing (i.e., the partial differential equation), its defining formula can be found by various methods, most based on integration. Important partial differential equations include the heat equation, the wave equation, and Laplace's equation, which are central to mathematical physics.
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Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function). Because the derivative is a rate of change, such an equation states how a function changes but does not specify the function itself. Given sufficient initial conditions, however, such as a specific function value, the function can be found by various methods, most based on integration.
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Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. Differential equations are very common in physics, engineering, and all fields involving quantitative study of change. They are used whenever a rate of change is known but the process giving rise to it is not. The solution of a differential equation is generally a function whose derivatives satisfy the equation. Differential equations are classified into several broad categories. The most important are ordinary differential equations (ODEs), in which change depends on a single variable, and partial differential equations (PDEs), in which change depends on several variables. Seealso differentiation.
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A simple example is Newton's second law of motion, which leads to the differential equation
for the motion of a particle of mass m. In general, the force F depends upon the position of the particle x(t) at time t, and thus the unknown function x(t) appears on both sides of the differential equation, as is indicated in the notation F(x(t)).
Ordinary differential equations arise in many different contexts including geometry, mechanics, astronomy and population modelling. Many famous mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler.
Much study has been devoted to the solution of ordinary differential equations. In the case where the equation is linear, it can be solved by analytical methods. Unfortunately, most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions are arrived at using computer approximations (see numerical ordinary differential equations).
Let y be an unknown function
in x with the nth derivative of y, then an equation of the form
is called an ordinary differential equation (ODE) of order n; for vector valued functions,
When a differential equation of order n has the form
A differential equation not depending on x is called autonomous.
with ai(x) and r(x) continuous functions in x. The function r(x) is called the source term; if r(x)=0 then the linear differential equation is called homogeneous, otherwise it is called non-homogeneous or inhomogeneous.
Given a differential equation
is called the solution or integral curve for F, if u is n-times differentiable on I, F is defined for all
Given two solutions
u is called an extension of v if I ⊂ J and
A solution which has no extension is called a global solution.
A general solution of an n-th order equation is a solution containing n arbitrary variables, corresponding to n constants of integration. A particular solution is derived from the general solution by setting the constants to particular values, often chosen to fulfill set 'initial conditions or boundary conditions'. A singular solution is a solution that can't be derived from the general solution.
Any differential equation of order n can be written as a system of n first-order differential equations. Given an explicit ordinary differential equation of order n and dimension 1,
we define a new family of unknown functions
We can then rewrite the original differential equation as a system of differential equations with order 1 and dimension n.
which can be written concisely in vector notation as
A well understood particular class of differential equations is linear differential equations. We can always reduce an explicit linear differential equation of any order to a system of differential equation of order 1
which we can write concisely using matrix and vector notation as
The set of solutions for a system of homogeneous linear differential equations of order 1 and dimension n
forms an n-dimensional vector space. Given a basis for this vector space , which is called a fundamental system, every solution can be written as
The n × n matrix
is called fundamental matrix. In general there is no method to explicitly construct a fundamental system, but if one solution is known d'Alembert reduction can be used to reduce the dimension of the differential equation by one.
The set of solutions for a system of inhomogeneous linear differential equations of order 1 and dimension n
can be constructed by finding the fundamental system to the corresponding homogeneous equation and one particular solution to the inhomogeneous equation. Every solution to nonhomogeneous equation can then be written as
If a system of homogeneous linear differential equations has constant coefficients
then we can explicitly construct a fundamental system. The fundamental system can be written as a matrix differential equation
with solution as a matrix exponential
which is a fundamental matrix for the original differential equation. To explicitly calculate this expression we first transform A into Jordan normal form
and then evaluate the Jordan blocks
Sturm-Liouville theory is a general method for resolution of second order linear equations with variable coefficients.
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