Search references for FORCING FUNCTION-DIFFERENTIAL-EQUATIONS. Phrases containing FORCING FUNCTION-DIFFERENTIAL-EQUATIONS
See searches and references containing FORCING FUNCTION-DIFFERENTIAL-EQUATIONS!FORCING FUNCTION-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)
Differential equation containing derivatives with respect to only one variable
more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations (PDEs)
Ordinary differential equation
Ordinary_differential_equation
Recurrence relation Matrix difference equation Rational difference equation Examples of differential equations Autonomous system (mathematics) Picard–Lindelöf
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
Topics referred to by the same term
Forcing function can mean: In differential calculus, a function that appears in the equations and is only a function of time, and not of any of the other
Forcing_function
Differential equation important in physics
(2010). Partial Differential Equations. Providence (R.I.): American Mathematical Soc. ISBN 978-0-8218-4974-3. "Linear Wave Equations", EqWorld: The World
Wave_equation
Special function in the physical sciences
Frank William John (1974). "Differential Equations with a Parameter : Turning Points". Asymptotics and Special Functions. New York: Academic Press. pp
Airy_function
Second-order partial differential equation
partial differential equations. Laplace's equation is also a special case of the Helmholtz equation. The general theory of solutions to Laplace's equation is
Laplace's_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
Partial differential equation
the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the position or
Fokker–Planck_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
Differential equations involving stochastic processes
stochastic differential equations. Stochastic differential equations can also be extended to differential manifolds. Stochastic differential equations originated
Stochastic differential equation
Stochastic_differential_equation
Equations that describe the behavior of a physical system
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically
Equations_of_motion
Non-linear stochastic partial differential equation
mathematics, the Kardar–Parisi–Zhang (KPZ) equation is a non-linear stochastic partial differential equation, introduced by Mehran Kardar, Giorgio Parisi
Kardar–Parisi–Zhang_equation
Second-order partial differential equation describing motion of mechanical system
classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of
Euler–Lagrange_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
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
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
In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form d x d t ( t ) ∈ F ( t , x (
Differential_inclusion
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
Equations of motion for viscous fluids
Navier–Stokes equations (/nævˈjeɪ ˈstoʊks/ nav-YAY STOHKS) describe the motion of viscous fluids. This system of partial differential equations was named
Navier–Stokes_equations
Study of rates of change
used to find the maxima and minima of functions. Equations involving derivatives are called differential equations and are fundamental in describing natural
Differential_calculus
equation, a first order nonlinear ordinary differential equation Euler conservation equations, a set of quasilinear first-order hyperbolic equations used
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
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
Stochastic differential equation
In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination
Langevin_equation
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
Polynomial sequence
second-order differential equations are in fact linear combinations of both Hermite polynomials and confluent hypergeometric functions of the first kind
Hermite_polynomials
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
System of complete and orthogonal polynomials
settings, Legendre's differential equation arises naturally whenever one solves Laplace's equation (and related partial differential equations) by separation
Legendre_polynomials
General form of cryptanalysis applicable primarily to block ciphers
work to determine the key as simply brute forcing the key. The AES non-linear function has a maximum differential probability of 4/256 (most entries however
Differential_cryptanalysis
Non-linear partial differential equation encountered in problems of wave propagation
, then equation (2) becomes (1). Eikonal equations naturally arise in the WKB method and the study of Maxwell's equations. Eikonal equations provide
Eikonal_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
Branch of mathematics
It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives and are ubiquitous in
Calculus
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)
Formulation of classical mechanics
that the Euler–Lagrange equations form a n × n {\displaystyle n\times n} system of second-order ordinary differential equations. Inverting the matrix H
Hamilton–Jacobi_equation
Method to solve constrained optimization problems
local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly
Lagrange_multiplier
Mathematics of smooth surfaces
Differential Equations II: Qualitative Studies of Linear Equations, Springer-Verlag, ISBN 978-1-4419-7051-0 Taylor, Michael E. (1996b), Partial Differential Equations
Differential geometry of surfaces
Differential_geometry_of_surfaces
Non-linear second order differential equation and its attractor
The Duffing equation (or Duffing oscillator), named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model
Duffing_equation
Finding values for variables that make an equation true
is {√2, −√2}. When an equation contains several unknowns, and when one has several equations with more unknowns than equations, the solution set is often
Equation_solving
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
Thermodynamic quantity
quantity dY is an exact differential, while δX is not, it is an inexact differential. Infinitesimal changes in a process function may be integrated, but
Process_function
Type of differential operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined
Elliptic_operator
Equation of statistical mechanics
convection–diffusion equation. The equation is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in
Boltzmann_equation
Generalization of the Dirac equation
}-m)\Psi .} Dirac equation in the algebra of physical space Dirac spinor Maxwell's equations in curved spacetime Two-body Dirac equations Lawrie, Ian D.
Dirac equation in curved spacetime
Dirac_equation_in_curved_spacetime
Function specifying the behavior of a component in an electronic or control system
response; it corresponds to the homogeneous solution of the differential equation. The transfer function for an LTI system may be written as the product: H (
Transfer_function
Nonlinear partial differential equation
The sine-Gordon equation is a second-order nonlinear partial differential equation for a function φ {\displaystyle \varphi } dependent on two variables
Sine-Gordon_equation
Generalized function whose value is zero everywhere except at zero
delta function arise as fundamental solutions or Green's functions to physically motivated elliptic or parabolic partial differential equations. In the
Dirac_delta_function
Describing pressure difference over an interface in fluid mechanics
Young–Laplace–Gauss equation, as Carl Friedrich Gauss unified the work of Young and Laplace in 1830, deriving both the differential equation and boundary conditions
Young–Laplace_equation
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
Differential equation exhibiting high rate of dissipation
corresponds to using an "implicit" method for differential equations. Another example is in solving nonlinear equations of the form x = g ( x ) {\displaystyle
Stiff_equation
Relativistic wave equation in quantum mechanics
where the equation describes the dynamics of spin-0 fields. Mathematically, it is a linear second-order hyperbolic partial differential equation that is
Klein–Gordon_equation
Branch of mathematical analysis
mathematics. Fractional differential equations, also known as extraordinary differential equations, are a generalization of differential equations through the application
Fractional_calculus
Description of the time-evolution of plasma
plasma physics, the Vlasov equation is a differential equation describing the time evolution of the distribution function of a collisionless plasma consisting
Vlasov_equation
Quasilinear first-order ordinary differential equation
classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid
Euler's equations (rigid body dynamics)
Euler's_equations_(rigid_body_dynamics)
Specific mathematical differential form
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
Inexact_differential
or "master equations" are: The four most common Maxwell's relations are: More relations include the following. Other differential equations are: U = N
Table of thermodynamic equations
Table_of_thermodynamic_equations
Hydrodynamic formulation of the Schrödinger equations
the Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's alternative formulation of the Schrödinger equation for a spinless
Madelung_equations
Circulation density in a vector field
spaces. In 3 dimensions, a differential 0-form is a real-valued function f ( x , y , z ) {\displaystyle f(x,y,z)} ; a differential 1-form is the following
Curl_(mathematics)
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
Physical system that responds to a restoring force proportional to displacement
t^{2}}}=m{\ddot {x}}=-kx.} Solving this differential equation, we find that the motion is described by the function x ( t ) = A sin ( ω t + φ ) , {\displaystyle
Harmonic_oscillator
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
Mathematical simplification technique in physical sciences
of dynamical systems and differential equations topics List of partial differential equation topics Differential equations of mathematical physics Although
Nondimensionalization
Scalar physical quantities representing system states
first and second laws of thermodynamics, a set of differential equations known as the fundamental equations follow. (Actually they are all expressions of
Thermodynamic_potential
Equations describing behavior of a model
governing differential equations within biology is Lotka-Volterra equations are prey-predator equations A governing equation may also be a state equation, an
Governing_equation
Equations of fluid dynamics
the equations, such as Navier–Stokes existence and smoothness, is one of the important unsolved problems in mathematics. The Navier–Stokes equations are
Derivation of the Navier–Stokes equations
Derivation_of_the_Navier–Stokes_equations
Equations in thermodynamics
commonly called "the equation of state" is just one of many possible equations of state.) If we know all k+2 of the above equations of state, we may reconstitute
Thermodynamic_equations
Formulation of classical mechanics
This constraint allows the calculation of the equations of motion of the system using Lagrange's equations. Newton's laws and the concept of forces are
Lagrangian_mechanics
Assignment of a vector to each point in a subset of Euclidean space
direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. The elements of differential and integral
Vector_field
Equation giving the form of a central force
nonlinear, ordinary differential equation. A unique solution is impossible in the case of circular motion about the center of force. The shape of an orbit
Binet_equation
Partial differential equation
In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation ∂ w ∂ z ¯ = μ ∂ w ∂ z . {\displaystyle {\frac
Beltrami_equation
Second-order control system
space under the effect of a time-varying force input u {\displaystyle {\textbf {u}}} . The differential equations which represent a double integrator are:
Double_integrator
}{=}}\ e^{-\zeta /2}T(\tau )P(\zeta )} , we obtain two ordinary differential equations coupled by a constant β {\displaystyle \beta } d T d τ + β T = 0
Mason–Weaver_equation
Mathematical model of the time dependence of a point in space
formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy x ˙ −
Dynamical_system
Equation
The Cauchy momentum equation is a vector partial differential equation put forth by Augustin-Louis Cauchy that describes the non-relativistic momentum
Cauchy_momentum_equation
Formulae for relative strengths of military forces
Lanchester equations are differential equations describing the time dependence of two armies' strengths A and B as a function of time, with the function depending
Lanchester's_laws
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
Type of simple planetary gear train
When this function is not required, the differential can be "unlocked" to function as a regular open differential. Locking differentials are mostly used
Differential (mechanical device)
Differential_(mechanical_device)
Mathematical description of quantum state
wave function resembles the Pauli wave function for the electron. Later, other relativistic wave equations were found. All these wave equations are of
Wave_function
Fundamental mechanical principles
Lanczos Action principles are applied to derive differential equations like the Euler–Lagrange equations or as direct applications to physical problems
Action_principles
Methods of mathematical approximation
of equations" D {\displaystyle D} include algebraic equations, differential equations (e.g., the equations of motion and commonly wave equations), thermodynamic
Perturbation_theory
Method for load calculation in construction
_{A}z\sigma _{xx}~\mathrm {d} A} To close the system of equations we need the constitutive equations that relate stresses to strains (and hence stresses to
Euler–Bernoulli_beam_theory
Equilibrium equation
equation takes the same form as the Hicks equation from fluid dynamics. This equation is a two-dimensional, nonlinear, elliptic partial differential equation
Grad–Shafranov_equation
Electromagnetism in general relativity
Cartesian) coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Physical quantity of dimension energy × time
small perturbations is equivalent to a set of differential equations (called the Euler–Lagrange equations) that may be obtained using the calculus of variations
Action_(physics)
Property of differential equations describing physical phenomena
study of analytic partial differential equations. Surprisingly, the theorem does not hold in the setting of smooth functions; an example discovered by
Well-posed_problem
Equations to approximate global atmospheric flow
The primitive equations are a set of nonlinear partial differential equations that are used to approximate global atmospheric flow and are used in most
Primitive_equations
Millennium Prize Problem
Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
Method for solving certain nonlinear partial differential equations
linear partial differential equations. Using a pair of differential operators, a 3-step algorithm may solve nonlinear differential equations; the initial
Inverse_scattering_transform
Branch of mathematics
the study of differential equations for connections on bundles, and the resulting geometric moduli spaces of solutions to these equations as well as the
Differential_geometry
Computational method for solving partial differential equations
The Kansa method is a computer method used to solve partial differential equations. Its main advantage is it is very easy to understand and program on
Kansa_method
Mathematical approach to quantum physics
\langle n|} on the left, this can be reduced to a set of coupled differential equations for the amplitudes, d c n d t = − i ℏ ∑ k ⟨ n | V ( t ) | k ⟩ c
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Overview of mechanics based on the least action principle
a set of differential equations. A problem is regarded as solved when the particles coordinates at time t are expressed as simple functions of t and of
Analytical_mechanics
Complex vector of electromagnetic fields
Heinrich Martin Weber published the fourth edition of "The partial differential equations of mathematical physics according to Riemann's lectures" in two
Riemann–Silberstein_vector
Physical field theory with no forces/interactions
physics, a free field is a field whose equations of motion are given by linear partial differential equations. Such linear PDE's have a unique solution
Free_field
Characteristic time in a system
their inputs. An example solution to the differential equation with initial value V0 and no forcing function is V ( t ) = V 0 e − t / τ {\displaystyle
Time_constant
Physical theory describing classical fields
both will vary in time. They are determined by Maxwell's equations, a set of differential equations which directly relate E and B to the electric charge density
Classical_field_theory
Two-dimensional laminar boundary layer that forms on a semi-infinite plate
edge of the plate). This leads to a reduced set of equations known as the boundary layer equations. For steady incompressible flow with constant viscosity
Blasius_boundary_layer
dynamics, Hicks equation, sometimes also referred as Bragg–Hawthorne equation or Squire–Long equation, is a partial differential equation that describes
Hicks_equation
Branch of mathematics
calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were
Mathematical_analysis
Foundational law of electromagnetism relating electric field and charge distributions
Gauss's flux theorem or sometimes Gauss's theorem, is one of Maxwell's equations. It is an application of the divergence theorem, and it relates the distribution
Gauss's_law
1865 physics paper by James Maxwell
entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations which were to become known as Maxwell's equations, until this
A Dynamical Theory of the Electromagnetic Field
A_Dynamical_Theory_of_the_Electromagnetic_Field
FORCING FUNCTION-DIFFERENTIAL-EQUATIONS
FORCING FUNCTION-DIFFERENTIAL-EQUATIONS
Surname or Lastname
English
English : ethnic name from Old French Lohereng ‘man from Lorraine’ (see Lorraine).
Surname or Lastname
English (mainly Lancashire and Cheshire)
English (mainly Lancashire and Cheshire) : unexplained.Probably an altered form of German Dornig, which is probably a nickname for someone with a sharp tongue, from an adjectival derivative of Middle High German, Middle Low German dorn ‘thorn’. The suffixes -ig and -ing were often interchanged in Pennsylvania German and elsewhere. The name may also refer to a sloe bush.
Girl/Female
Afghan, Arabic, Australian, Indian, Muslim
Fiction; Romance; Story
Girl/Female
English Latin
Girl/Female
American, British, English, Latin
Farmer; Variant of Georgina
Boy/Male
German French
Famous in battle.
Surname or Lastname
English
English : habitational name from a place in West Sussex, so named from the Old English personal name Fēra + -ingas ‘people of’, ‘family of’, or ‘followers of’.
Surname or Lastname
English
English : habitational name from places in Oxfordshire and West Sussex named Goring, from Old English GÄringas ‘people of GÄra’, a short form of the various compound names with the first element gÄr ‘spear’.German (Göring) : see Goering.
Boy/Male
Indian
Friction
Boy/Male
Afghan, Arabic, Muslim, Pashtun
One who can Differentiate; Comely; One who Distinguishes Truth from Falsehood
Surname or Lastname
English (Somerset)
English (Somerset) : unexplained.
Boy/Male
French, German
Renowned Warrior's Son; From Lorraine; Son of the Famous Warrior
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Girl/Female
Tamil
Ankshika | அஂகà¯à®·à¯€à®•à®¾
It’s derived from the root word - anksh that means a fraction. Ankshika means the fraction of the cosmos
Ankshika | அஂகà¯à®·à¯€à®•à®¾
Boy/Male
Latin
Strong; fortunate.
Girl/Female
Hindu, Indian
Fraction of the Cosmos
Surname or Lastname
English
English : from a diminutive of Moore 2, 3.North German (Möring) : patronymic from the nickname Mohr (see Mohr 2).North German (Möring) : habitational name from Möringen or Möhringen near Stendal and Stettin.Dutch : variant of Morin.
Boy/Male
German
Renowned Warrior's Son
Girl/Female
Bengali, Indian
Fraction of Time
Surname or Lastname
English
English : patronymic from Dear 1.German (Döring) : see Doering.
FORCING FUNCTION-DIFFERENTIAL-EQUATIONS
FORCING FUNCTION-DIFFERENTIAL-EQUATIONS
Biblical
blackness; heat
Boy/Male
Hindu, Indian, Marathi
Given by Yajna
Girl/Female
Indian
The beautiful woman, Beautiful jewel
Surname or Lastname
English
English : perhaps a variant of Dear.German (Därr) : from a short form of a Germanic personal name, perhaps related to Old High German dart ‘spear’.Variant spelling of German Dorr.
Girl/Female
Hindu
Trees and creppers over grown, Arbour
Boy/Male
Tamil
Gudakesha | கà¯à®¤à®¾à®•à¯‡à®·à®¾
The archer Arjuna
Surname or Lastname
English (Norfolk)
English (Norfolk) : unexplained. It may be a result of misdivision of some personal name ending in -n + the surname Hudd (see Hutt).
Boy/Male
Indian
One who is limitless and endless
Girl/Female
Arabic, Muslim
Always Listening; World
Girl/Female
Tamil
Valour, Strength
FORCING FUNCTION-DIFFERENTIAL-EQUATIONS
FORCING FUNCTION-DIFFERENTIAL-EQUATIONS
FORCING FUNCTION-DIFFERENTIAL-EQUATIONS
FORCING FUNCTION-DIFFERENTIAL-EQUATIONS
FORCING FUNCTION-DIFFERENTIAL-EQUATIONS
a.
Pertaining to, or connected with, a function or duty; official.
a.
Pertaining to the function of an organ or part, or to the functions in general.
v. t.
To obtain the differential, or differential coefficient, of; as, to differentiate an algebraic expression, or an equation.
n.
The act of joining, or the state of being joined; union; combination; coalition; as, the junction of two armies or detachments; the junction of paths.
n.
The place or point of union, meeting, or junction; specifically, the place where two or more lines of railway meet or cross.
v. t.
To sell by auction.
v. t.
The act of uniting, or the state of being united; junction.
a.
Of or pertaining to a differential, or to differentials.
a.
Relating to or indicating a difference; creating a difference; discriminating; special; as, differential characteristics; differential duties; a differential rate.
n.
See Furring.
n.
The eliminant of the n partial differentials of any homogenous function of n variables. See Eliminant.
v. t.
To give sanction to; to ratify; to confirm; to approve.
n.
The operation of finding the primitive function which has a given function for its differential coefficient. See Integral.
a.
Pertaining to agriculture; devoted to, adapted to, or engaged in, farming; as, farming tools; farming land; a farming community.
n.
The things sold by auction or put up to auction.
pl.
of Differentia
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
n.
A small difference in rates which competing railroad lines, in establishing a common tariff, allow one of their number to make, in order to get a fair share of the business. The lower rate is called a differential rate. Differentials are also sometimes granted to cities.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
v. t.
To distinguish or mark by a specific difference; to effect a difference in, as regards classification; to develop differential characteristics in; to specialize; to desynonymize.