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Differential equation that is linear with respect to the unknown function
In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written
Linear_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
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
Differential equation containing derivatives with respect to only one variable
with stochastic differential equations (SDEs) where the modeled process is random. A linear differential equation is a differential equation that is defined
Ordinary differential equation
Ordinary_differential_equation
Type of ordinary differential equation
differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations
Homogeneous differential equation
Homogeneous_differential_equation
System where changes of output are not proportional to changes of input
regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown
Nonlinear_system
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)
Type of ordinary differential equation
Bernoulli equation is the logistic differential equation. When n = 0 {\displaystyle n=0} , the differential equation is linear. When n = 1 {\displaystyle n=1}
Bernoulli differential equation
Bernoulli_differential_equation
Concept in differential equation mathematics
substantially different. More precisely, consider an ordinary linear differential equation of n-th order f ( n ) ( z ) + ∑ i = 0 n − 1 p i ( z ) f ( i )
Regular_singular_point
Mathematical formula expressing equality
the term partial differential equation, which may be with respect to more than one independent variable. Linear differential equations, which have solutions
Equation
The Fuchsian theory of linear differential equations, which is named after Lazarus Immanuel Fuchs, provides a characterization of various types of singularities
Fuchsian_theory
Identity relating to differential equations
or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential
Abel's_identity
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
Second order linear differential equation featuring a periodic function
In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation d 2 y d t 2 + f ( t ) y = 0
Hill_differential_equation
Group of differential equations
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 a system
System of differential equations
System_of_differential_equations
they are to solve compared to linear differential equations. This list presents nonlinear ordinary differential equations that have been named, sorted
List of nonlinear ordinary differential equations
List_of_nonlinear_ordinary_differential_equations
Class of differential equations expressible in differential algebra
mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several
Algebraic differential equation
Algebraic_differential_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
Principle of quantum mechanics
Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of
Quantum_superposition
Mechanical analogue computer to solve differential equations
The differential analyser is a mechanical analogue computer designed to solve differential equations by integration, using wheel-and-disc mechanisms to
Differential_analyser
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
Equation involving both integrals and derivatives of a function
an integro-differential equation is an equation that involves both integrals and derivatives of a function. The general first-order, linear (only with
Integro-differential_equation
Class of partial differential equations
In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are
Elliptic partial differential equation
Elliptic_partial_differential_equation
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
Mathematical relation defining a sequence
linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in
Linear recurrence with constant coefficients
Linear_recurrence_with_constant_coefficients
Sequence of differential equation solutions
Laguerre's differential equation: x y ″ + ( 1 − x ) y ′ + n y = 0 , y = L ( x ) {\displaystyle xy''+(1-x)y'+ny=0,\ y=L(x)} which is a second-order linear differential
Laguerre_polynomials
Properties of mathematical relationships
Linear actuator Linear element Linear foot Linear system Linear programming Linear differential equation Bilinear Multilinear Linear motor Linear interpolation
Linearity
named linear ordinary differential equations. List of nonlinear ordinary differential equations List of nonlinear partial differential equations List of
List of linear ordinary differential equations
List_of_linear_ordinary_differential_equations
Matrix operation generalizing exponentiation of scalar numbers
ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the
Matrix_exponential
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
Generalization of the hypergeometric differential equation
The equation is also known as the Papperitz equation. The hypergeometric differential equation is a second-order linear differential equation which
Riemann's differential equation
Riemann's_differential_equation
Determinant of the matrix of first derivatives of a set of functions
mathematician Józef Wroński, and is used in the study of differential equations, where it can show the linear independence of certain sets of solutions. The Wronskian
Wronskian
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
Sturm–Liouville_theory
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
Type of partial differential equations
mathematics, a hyperbolic partial differential equation of order n {\displaystyle n} is a partial differential equation (PDE) that, roughly speaking, has
Hyperbolic partial differential equation
Hyperbolic_partial_differential_equation
Study of differential field extensions induced by linear differential equations
theory says very roughly that a linear differential equation can be solved by quadratures if and only if its differential Galois group is connected and
Picard–Vessiot_theory
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
Special function in the physical sciences
{d^{2}y}{dx^{2}}}-xy=0,} known as the Airy equation or the Stokes equation. Because the solution of the linear differential equation d 2 y d x 2 − k y = 0 {\displaystyle
Airy_function
Algebraic equation on which the solution of a differential equation depends
differential 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)
Linear differential equation
An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect
Adjoint_equation
Function defined by a hypergeometric series
linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.
Hypergeometric_function
Branch of ordinary differential equations
is a branch of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form x ˙ = A ( t )
Floquet_theory
Description of a quantum-mechanical system
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery
Schrödinger_equation
Procedure for solving differential equations
method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to
Variation_of_parameters
Linear optimal control technique
The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called
Linear–quadratic_regulator
Mathematical concept
broad class of second order linear differential equations as follows. By standard changes of variables, a second order equation can often be changed to one
Stokes_phenomenon
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 ordinary differential equation
{dx}{dp}}+{\frac {xf'(p)+g'(p)}{f(p)-p}}=0} The above equation is a first order linear differential equation: d x d p + f ′ ( p ) f ( p ) − p x = − g ′ ( p )
D'Alembert's_equation
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
Second-order linear differential equation
Chebyshev's equation is the second order linear differential equation ( 1 − x 2 ) d 2 y d x 2 − x d y d x + p 2 y = 0 , {\displaystyle (1-x^{2}){d^{2}y
Chebyshev_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
dynamical system and differential equation topics. Deterministic system (mathematics) Linear system Partial differential equation Dynamical systems and
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
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
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
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
Statistical function that defines the quantiles of a probability distribution
characterized as solutions of non-linear ordinary and partial differential equations. The ordinary differential equations for the cases of the normal, Student
Quantile_function
Type of mathematical equation
{\displaystyle \mathbf {A} } is constant and has n linearly independent eigenvectors, this differential equation has the following general solution, x ( t )
Matrix_differential_equation
Solution method for linear differential equations
method is a technique for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for
WKB_approximation
Solvable form of differential equation
An inexact differential equation is a differential equation of the form: M ( x , y ) d x + N ( x , y ) d y = 0 {\displaystyle M(x,y)\,dx+N(x,y)\,dy=0}
Inexact_differential_equation
Fundamental principle of physics
principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli
Superposition_principle
Exponential representation for differential equations
homogeneous linear differential equation for a linear operator. In particular, it furnishes the fundamental matrix of a system of linear ordinary differential equations
Magnus_expansion
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
Family of solutions to related differential equations
rather than solutions to differential equations. Because the differential equation is second-order, there must be two linearly independent solutions: one
Bessel_function
Mathematical function often applied to matrices
square matrices and bounded linear operators. The original purpose was to estimate solutions to linear differential equations x ˙ = A x + r {\displaystyle
Logarithmic_norm
Class of second-order linear partial differential equations
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent
Parabolic partial differential equation
Parabolic_partial_differential_equation
Method for solving differential equations
solution into the differential equation to find a recurrence relation for the coefficients. Consider the second-order linear differential equation a 2 ( z ) f
Power series solution of differential equations
Power_series_solution_of_differential_equations
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
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
Criterion for integration in terms of elementary functions
der Put, Marius; Singer, Michael F. (2003), Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental
Liouville's theorem (differential algebra)
Liouville's_theorem_(differential_algebra)
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
Algebraic study of differential equations
mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators
Differential_algebra
Pattern defining an infinite sequence of numbers
be calculated by repeatedly applying the equation. In linear recurrences, the nth term is equated to a linear function of the k {\displaystyle k} previous
Recurrence_relation
Algebraic structure in linear algebra
c} ) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation D ( f ) = 0 {\displaystyle
Vector_space
Partial differential equation describing the evolution of temperature in a region
specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier
Heat_equation
Constant solution to a differential equation
mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. The point x ~ ∈ R n {\displaystyle
Equilibrium point (mathematics)
Equilibrium_point_(mathematics)
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 function, denoted exp(x) or e^x
antiderivative of its argument. More generally, the solutions of every linear differential equation with constant coefficients can be expressed in terms of exponential
Exponential_function
Physical system satisfying the superposition principle
definition of a linear system is analogous to the definition of a linear differential equation in calculus, and a linear transformation in linear algebra. A
Linear_system
Method for solving certain nonlinear partial differential equations
a nonlinear partial differential equation to solving 2 linear ordinary differential equations and an ordinary integral equation, a method ultimately
Inverse_scattering_transform
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
Mathematical theorem
second order linear differential equations. Basically the theorem states that given two linear independent solutions of such an equation the zeros of
Sturm_separation_theorem
Study of rates of change
function at a point generally determines the best linear approximation to the function at that point. Differential calculus is one of the two traditional divisions
Differential_calculus
Type of differential equation
of the 'leak'. The equation is a nonhomogeneous first-order linear differential equation. For constant C its solution is x ( t ) = k e − A t + C A {\displaystyle
Leaky_integrator
Set of equations to be solved together
System of linear equations System of nonlinear equations System of bilinear equations System of polynomial equations System of differential equations System
System_of_equations
Growth curve model
L_{\infty }} is asymptotic size. It is the solution of the following linear differential equation: d L d a = k ( L ∞ − L ) {\displaystyle {\frac {dL}{da}}=k(L_{\infty
Von_Bertalanffy_function
Describes state evolution of a linear system
matrix is used to find the general solution to the homogeneous linear differential equation x ˙ ( t ) = A ( t ) x ( t ) {\displaystyle {\dot {\mathbf {x}
State-transition_matrix
Control theory for nonlinear or time-variant systems
Linear control theory applies to systems made of devices which obey the superposition principle. They are governed by linear differential equations.
Nonlinear_control
Multiplicative factor in a mathematical expression
term rather than a constant coefficient. In particular, in a linear differential equation with constant coefficient, the constant coefficient term is generally
Coefficient
Equation that describes density changes of a material that is diffusing in a medium
differential operator del. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear. The equation above
Diffusion_equation
Integral transform useful in probability theory, physics, and engineering
tool for solving linear differential equations and dynamical systems by replacing ordinary differential equations and integral equations with algebraic
Laplace_transform
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
Combination of the diffusion and convection (advection) equations
convection–diffusion equation is a parabolic partial differential equation that combines the diffusion and convection (advection) equations. It describes physical
Convection–diffusion_equation
Criteria for the oscillation and non-oscillation of some linear differential equations
solutions of some linear differential equations in the real domain. It extends the Picone identity from two to three or more differential equations of the second
Mingarelli_identity
Function specifying the behavior of a component in an electronic or control system
{Y(z)}{X(z)}}={\frac {{\mathcal {Z}}\{y[n]\}}{{\mathcal {Z}}\{x[n]\}}}.} A linear differential equation with constant coefficients L [ u ] = d n u d t n + a 1 d n −
Transfer_function
certain linear differential equations, so called Fuchsian equations. It is named after Lazarus Immanuel Fuchs. A linear differential equation in which
Fuchs_relation
using a solution of a linear differential equation. Pseudospherical surfaces can be described as solutions of the sine-Gordon equation, and hence the Bäcklund
Bäcklund_transform
Family of second-order differential equations
of dynamical systems and differential equations, a Liénard equation is a type of second-order ordinary differential equation named after the French physicist
Liénard_equation
Method for solving linear differential equations using the Laplace transform
s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions. First consider the following
Laplace transform applied to differential equations
Laplace_transform_applied_to_differential_equations
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
Initial estimate or framework to the solution of a mathematical problem
solution of a homogeneous linear differential equation to take an exponential form, or a power form in the case of a difference equation. More generally, one
Ansatz
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