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Type of ordinary differential equation
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written
Homogeneous differential equation
Homogeneous_differential_equation
Differential equation containing derivatives with respect to only one variable
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other
Ordinary differential equation
Ordinary_differential_equation
Differential equation that is linear with respect to the unknown function
the equation are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations have
Linear_differential_equation
Algebraic equation on which the solution of a differential equation depends
equation or difference equation. The characteristic equation can only be formed when the differential equation is linear and homogeneous, and has constant
Characteristic equation (calculus)
Characteristic_equation_(calculus)
Type of differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives
Partial_differential_equation
Type of functional equation (mathematics)
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions
Differential_equation
Procedure for solving differential equations
solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions
Variation_of_parameters
Identity relating to differential equations
Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear
Abel's_identity
Second-order partial differential equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its
Laplace's_equation
Partial differential equation describing the evolution of temperature in a region
The heat equation is a parabolic partial differential equation that occurs in the theory of heat transfer and models the evolution of the temperature
Heat_equation
Methods used to find numerical solutions of ordinary differential equations
for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their
Numerical methods for ordinary differential equations
Numerical_methods_for_ordinary_differential_equations
Differential equations involving stochastic processes
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution
Stochastic differential equation
Stochastic_differential_equation
System where changes of output are not proportional to changes of input
system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear
Nonlinear_system
Ordinary differential equation
Euler–Cauchy equation, also known as a Cauchy–Euler equation, equidimensional equation, or Euler's equation, is a linear ordinary differential equation for which
Cauchy–Euler_equation
Method of solution for inhomogeneous ODEs
it only works for differential equations that follow certain forms. Consider a linear non-homogeneous ordinary differential equation of the form ∑ i =
Method of undetermined coefficients
Method_of_undetermined_coefficients
Function with a multiplicative scaling behaviour
are exactly the solution of a specific partial differential equation. More precisely: Euler's homogeneous function theorem—If f is a (partial) function
Homogeneous_function
Differential equation exhibiting high rate of dissipation
computations, stiff equations are invariably solved using adaptive methods. There is a rich literature on stiff differential equations, but intuitive descriptions
Stiff_equation
Equations with an unknown function under an integral sign
integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals
Integral_equation
Equations describing classical electromagnetism
Maxwell's equations are a set of coupled partial differential equations that describe how electric and magnetic fields are generated by electric charges
Maxwell's_equations
Topics referred to by the same term
Homogeneous system: Homogeneous system of linear algebraic equations System of homogeneous differential equations System of homogeneous first-order differential
Homogeneous_system
equation Hypergeometric differential equation Jimbo–Miwa–Ueno isomonodromy equations Painlevé equations Picard–Fuchs equation to describe the periods
List of named differential equations
List_of_named_differential_equations
Function that only depends on time
system of differential equations used to describe a time-dependent process, a forcing function is a function that appears in the equations and is only
Forcing function (differential equations)
Forcing_function_(differential_equations)
Technique for solving linear ordinary differential equations
technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y 1 ( x ) {\displaystyle y_{1}(x)}
Reduction_of_order
of a non-homogeneous linear ordinary differential equation of any order. The exponential response formula is applicable to non-homogeneous linear ordinary
Exponential_response_formula
Type of differential equation subject to a particular solution methodology
mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in
Exact_differential_equation
Partial differential equation used in physics
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium
Electromagnetic_wave_equation
Partial differential equations with random force terms and coefficients
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary
Stochastic partial differential equation
Stochastic_partial_differential_equation
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
System of equations in mathematics
a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or
Differential-algebraic system of equations
Differential-algebraic_system_of_equations
Equation in physics
the wave equations make the partial differential equations inhomogeneous, if the source terms are zero the equations reduce to the homogeneous electromagnetic
Inhomogeneous electromagnetic wave equation
Inhomogeneous_electromagnetic_wave_equation
Differential equation for the description of waves or standing wave
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves
Wave_equation
Type of mathematical equation
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
Matrix_differential_equation
Otherwise, Euler's equation may refer to a non-differential equation, as in these three cases: Euler–Lotka equation, a characteristic equation employed in mathematical
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Group of differential equations
In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such
System of differential equations
System_of_differential_equations
Mathematical nomenclature
mathematical equations, particularly linear simultaneous equations, differential equations and integral equations, the terminology homogeneous is often used
Sides_of_an_equation
Type of ordinary differential equation
In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form y ′ + P ( x ) y = Q ( x ) y n , {\displaystyle
Bernoulli differential equation
Bernoulli_differential_equation
Elliptic partial differential equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the
Poisson's_equation
Field-equations in general relativity
tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE are the components
Einstein_field_equations
Mathematical model of waves on a shallow water surface
In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow
Korteweg–De_Vries_equation
Study of differential field extensions induced by linear differential equations
differential equation, using the differential Galois group of the field extension. A major goal is to describe when the differential equation can be solved
Picard–Vessiot_theory
Coordinate system used in projective geometry
coordinates, a single point can be represented by infinitely many homogeneous coordinates. The equation of a line through the origin ( 0 , 0 ) {\displaystyle (0
Homogeneous_coordinates
Equations modelling predator–prey cycles
Lotka–Volterra equations, also known as the Lotka–Volterra predator–prey model, are a pair of first-order nonlinear differential equations, frequently used
Lotka–Volterra_equations
Expression in differential equations
is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms
Liouville's_formula
Mathematical theorem
with continuous right hand side. Consider an ordinary linear homogeneous differential equation of the form y ″ + q ( x ) y = 0 {\displaystyle y''+q(x)y=0}
Kneser's theorem (differential equations)
Kneser's_theorem_(differential_equations)
Matrix consisting of linearly independent solutions to a linear differential equation
mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations x ˙ ( t ) = A ( t ) x ( t ) {\displaystyle {\dot {\mathbf
Fundamental matrix (linear differential equation)
Fundamental_matrix_(linear_differential_equation)
Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world
List of nonlinear ordinary differential equations
List_of_nonlinear_ordinary_differential_equations
Properties of mathematical relationships
differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation. Linearity of a homogeneous differential
Linearity
Set of partial differential equations on fluid flow
The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the
Shallow_water_equations
and the relations among them. At any ordinary point of a homogeneous linear differential equation of order n {\displaystyle n} there exists a fundamental
Fuchsian_theory
Partial differential equation with nonlinear terms
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different
Nonlinear partial differential equation
Nonlinear_partial_differential_equation
Matrix operation generalizing exponentiation of scalar numbers
linear differential equations. (See also matrix differential equation.) Recall from earlier in this article that a homogeneous differential equation of the
Matrix_exponential
Topics referred to by the same term
partial differential equations with highly oscillatory coefficients Homogeneous coordinates, used in projective spaces Homogeneous differential equation Homogeneous
Homogeneity_(disambiguation)
Millennium Prize Problem
mathematics since the early 20th century. The equations are a system of partial differential equations that describe the motion of a fluid in space. Although
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
Class of partial differential equations
the mathematical field of differential equations, the ultrahyperbolic equation is a class of partial differential equation (PDE) first described by R
Ultrahyperbolic_equation
Method for solving partial differential equations
partial differential equations, Duhamel's principle is a general method for obtaining solutions to inhomogeneous linear evolution equations like the
Duhamel's_principle
Type of ordinary differential equation
In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form y ( x ) = x d y d x + f ( d y d x ) {\displaystyle
Clairaut's_equation
Type of differential equation
In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function
Riccati_equation
Type of differential equation
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time
Delay_differential_equation
Eigenvalue problem for the Laplace operator
the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: ∇ 2 f = − k 2
Helmholtz_equation
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
In fluid dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
Technique for solving differential equations
the solving of a given equation involving differentials. It is commonly used to solve non-exact ordinary differential equations, but is also used within
Integrating_factor
Motion of launched objects due to gravity
integration of the ordinary differential equation, for instance by applying a reduction to a first-order system. The equation to be solved is d d t ( x
Projectile_motion
Mathematical concept
the principal homogeneous space. It is a one-form defined on P satisfying an integrability condition known as the Maurer–Cartan equation. Using this integrability
Maurer–Cartan_form
Mathematical descriptions of transmission line voltage and current
The telegrapher's equations (or telegraph equations) are a set of two coupled, linear partial differential equations that model voltage and current along
Telegrapher's_equations
Mathematical solution
specifically partial differential equations (PDEs), d'Alembert's formula is the general solution to the one-dimensional wave equation: u t t − c 2 u x x
D'Alembert's_formula
Pattern defining an infinite sequence of numbers
difference equations with differential equations, the methods of resolution of differential equations may often be applied to difference equations and thus
Recurrence_relation
Class of ordinary differential equations
applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form d d x [ p ( x ) d y d x ] + q ( x ) y = − λ w ( x )
Sturm–Liouville_theory
Mathematical theorem
the location of roots of solutions of homogeneous second order linear differential equations, that is, equations of the form y ″ + p ( x ) y ′ + q ( x
Sturm_separation_theorem
Mathematical transform that expresses a function of time as a function of frequency
} . A set of eigenfunctions is found by noting that the homogeneous differential equation [ U ( 1 2 π d d x ) + U ( x ) ] ψ ( x ) = 0 {\displaystyle
Fourier_transform
Topics referred to by the same term
function, a smooth function that is a solution of a linear homogeneous differential equation with polynomial coefficients Holonomic brain theory, model
Holonomic
linear second-order homogeneous differential equation x y ″ + ( n + 1 ) y ′ = y . {\displaystyle xy''+(n+1)y'=y.\qquad } This equation is of generalized
Bessel–Clifford_function
Mathematics of smooth surfaces
curves on the surface which satisfy a certain second-order ordinary differential equation which is specified by the first fundamental form. They are very
Differential geometry of surfaces
Differential_geometry_of_surfaces
Technique for solving hyperbolic partial differential equations
partial differential equations. The method is to reduce a partial differential equation (PDE) to a family of ordinary differential equations (ODEs) along
Method_of_characteristics
Equation involving both integrals and derivatives of a function
In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. The general first-order, linear
Integro-differential_equation
Type of functions, in mathematical analysis
several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension
Holonomic_function
Type of mathematical model
reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form ∂ t q = D _ _ ∇ 2 q
Reaction–diffusion_system
Typically linear operator defined in terms of differentiation of functions
of homogeneous functions. (Euler's homogeneous function theorem) In writing, following common mathematical convention, the argument of a differential operator
Differential_operator
Study of Galois symmetry groups of differential fields
extension, then G is called an elementary differential extension . Consider the homogeneous linear differential equation for a 1 , ⋯ , a n ∈ F {\displaystyle
Differential_Galois_theory
Mathematical relation defining a sequence
lag between iterates. The equation is called homogeneous if b = 0 and nonhomogeneous if b ≠ 0. If the equation is homogeneous, the coefficients determine
Linear recurrence with constant coefficients
Linear_recurrence_with_constant_coefficients
Type of problem involving ODEs or PDEs
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution
Boundary_value_problem
the initial homogeneous differential equation with nonconstant coefficients is changed to a series of non-homogeneous differential equations with constant
Unified_framework
In mathematics, an abstract differential equation is a differential equation in which the unknown function and its derivatives take values in some generic
Abstract differential equation
Abstract_differential_equation
Formulation of classical mechanics using momenta
Hamilton's equations consist of 2n first-order differential equations, while Lagrange's equations consist of n second-order equations. Hamilton's equations usually
Hamiltonian_mechanics
(p_{2}(x)y^{\prime })^{\prime }+q_{2}(x)y=0} be two homogeneous linear second order differential equations in self-adjoint form with 0 < p 2 ( x ) ≤ p 1 (
Sturm–Picone comparison theorem
Sturm–Picone_comparison_theorem
Exponential representation for differential equations
representation of the product integral solution of a first-order homogeneous linear differential equation for a linear operator. In particular, it furnishes the
Magnus_expansion
Method for solving differential equations
residuals (MWR) are methods for solving differential equations. The solutions of these differential equations are assumed to be well approximated by a
Method of mean weighted residuals
Method_of_mean_weighted_residuals
Equation in physics
determined. In the homogeneous case (f=0), the screened Poisson equation is the same as the time-independent Klein–Gordon equation. In the inhomogeneous
Screened_Poisson_equation
Determinant of the matrix of first derivatives of a set of functions
homogeneous-linear ordinary differential equation y ( n ) + L y = 0 {\displaystyle y^{(n)}+Ly=0} (where L {\displaystyle L} is a linear differential operator
Wronskian
Method of solution to differential equations
{\displaystyle L} is a linear differential operator, then the Green's function G {\displaystyle G} is the solution of the equation L G = δ , {\displaystyle
Green's_function
Ordinary differential equation
mechanics, the Rayleigh–Plesset equation or Besant–Rayleigh–Plesset equation is a nonlinear ordinary differential equation which governs the dynamics of
Rayleigh–Plesset_equation
Equation from probability theory
{\displaystyle P(t)=P^{t}.\,} The differential form of the Chapman–Kolmogorov equation is a representation of the master equation associated with a time-continuous
Chapman–Kolmogorov_equation
Method for load calculation in construction
furthered theories and formulated the differential equation of motion of a vibrating beam. The Euler–Bernoulli equation describes the relationship between
Euler–Bernoulli_beam_theory
Coordinates used to specify position of a line
tangential equation of the point. Similarly, for a point (x, y, z) given in homogeneous coordinates, the equation of the point in homogeneous tangential
Line_coordinates
Family of implicit and explicit iterative methods
method, used in discretization for the approximate solutions of differential equations. These methods were developed around 1900 by the German mathematicians
Runge–Kutta_methods
Existence and uniqueness of solutions to initial value problems
ordinary differential equations for y(t). Both differential equations will possess a single stationary point y = 0. First, the homogeneous linear equation dy/dt
Picard–Lindelöf_theorem
Plot of a dynamical system's trajectories in phase space
portrait represents the directional behavior of a system of ordinary differential equations (ODEs). The phase portrait can indicate the stability of the system
Phase_portrait
Method for solving differential equations
series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown
Power series solution of differential equations
Power_series_solution_of_differential_equations
Special mathematical functions defined on the surface of a sphere
surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics contains
Spherical_harmonics
Mathematical relationship describing the flow of groundwater through an aquifer
known in other fields as the diffusion equation or heat equation, it is a parabolic partial differential equation (PDE). This mathematical statement indicates
Groundwater_flow_equation
Group analysis of differential equations is a branch of mathematics that studies the symmetry properties of differential equations with respect to various
Group analysis of differential equations
Group_analysis_of_differential_equations
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HOMOGENEOUS DIFFERENTIAL-EQUATION
HOMOGENEOUS DIFFERENTIAL-EQUATION
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HOMOGENEOUS DIFFERENTIAL-EQUATION
HOMOGENEOUS DIFFERENTIAL-EQUATION
HOMOGENEOUS DIFFERENTIAL-EQUATION
HOMOGENEOUS DIFFERENTIAL-EQUATION
HOMOGENEOUS DIFFERENTIAL-EQUATION
HOMOGENEOUS DIFFERENTIAL-EQUATION
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