"미분방정식"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
(피타고라스님이 이 페이지를 개설하였습니다.)
 
 
(사용자 2명의 중간 판 58개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 +
==개요==
  
 +
* 미분방정식은 자연현상을 기술하는 수학적인 언어
 +
* 함수를 계수로 하여 미지수가 되는 일변수 함수와 고계도함수 사이에 만족되는 방정식을 말함
 +
* 학부과정에서는 [[상미분방정식]] 과목과 [[편미분방정식]]이 있음
 +
* 미분방정식의 해를 적당한 클래스의 함수(가령 초등함수, 초등함수의 적분) 들을 이용하여 표현하는 문제(solvability, integrability, quadrature)
 +
*  분류법
 +
** 미분방정식의 계(order)
 +
** 선형미분방정식과 비선형미분방정식
 +
** 상미분방정식과 편미분방정식
 +
 +
 +
 +
 +
 +
==일계 미분방정식==
 +
 +
* [[일계 선형미분방정식|일계선형미분방정식]]:<math>\frac{dy}{dt}+a(t)y=b(t)</math>
 +
* [[완전미분방정식]]:<math>M_y=N_x</math>를 만족시키는 <math>M(x, y)\, dx + N(x, y)\, dy = 0</math>  꼴의 미분방정식
 +
* 다음 미분방정식들은 비선형이다
 +
* [[리카티 미분방정식]]:<math>y' = A(x)+ B(x)y + C(x)y^2, A(x)\neq 0, C(x)\neq 0</math>
 +
* [[베르누이 미분방정식]]:<math>y'+ P(x)y = Q(x)y^n</math>
 +
 +
 +
 +
 +
 +
==이계 선형미분방정식==
 +
 +
*  다음 형태로 주어지는 미분방정식을 [[이계 선형 미분방정식|이계선형미분방정식]]이라 함:<math>\frac{d^2y}{dx^2}+p(x)\frac{dy}{dx}+q(x)y=g(x)</math>
 +
* [[상수계수 이계 선형미분방정식]]:<math>ay''+by'+cy=0</math>
 +
* [[에어리 (Airy) 함수와 미분방정식|Airy 미분방정식]]:<math>y'' - xy = 0</math>
 +
* [[베셀 미분방정식]]:<math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0</math>
 +
* [[에르미트 다항식(Hermite polynomials)]]:<math>y''-2xy'+\lambda y=0</math>
 +
* [[르장드르 다항식]]:<math>(1-x^2)y''-2xy'+\lambda(\lambda+1) y=0</math>
 +
* [[체비셰프 다항식]]:<math>(1-x^2)y''-xy'+\lambda^2 y=0</math>
 +
*  라게르 미분방정식:<math>xy''+(1-x)y'+\lambda y=0</math>
 +
* [[오일러 미분방정식]]:<math>x^2\frac{d^2y}{dx^2}+\alpha x\frac{dy}{dx}+\beta y=0</math>
 +
* [[초기하 미분방정식(Hypergeometric differential equations)]]:<math>z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0</math>
 +
* [[리만 미분방정식]]:<math>\frac{d^2w}{dz^2} + \left[ \frac{1-\alpha-\alpha'}{z-a} + \frac{1-\beta-\beta'}{z-b} + \frac{1-\gamma-\gamma'}{z-c} \right] \frac{dw}{dz}+\left[ \frac{\alpha\alpha' (a-b)(a-c)} {z-a} +\frac{\beta\beta' (b-c)(b-a)} {z-b} +\frac{\gamma\gamma' (c-a)(c-b)} {z-c} \right] \frac{w}{(z-a)(z-b)(z-c)}=0</math> 여기서 <math>\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1</math>
 +
 +
* [[호인 미분방정식(Heun's equation)]]:<math>\frac {d^2w}{dz^2} +  \left[\frac{\gamma}{z}+ \frac{\delta}{z-1} + \frac{\epsilon}{z-d} \right]  \frac {dw}{dz}  + \frac {\alpha \beta z -q} {z(z-1)(z-d)} w = 0</math> (여기서 <math>\epsilon=\alpha+\beta-\gamma-\delta+1</math>)
 +
 +
 +
 +
 +
 +
==비선형 미분방저식==
 +
 +
* [[팽르베 미분방정식(Painlevé Equations)|팽르베 미분방정식]]
 +
* [[바이어슈트라스 타원함수 ℘|바이어슈트라스의 타원함수]]:<math>(\frac{dw}{dz})^2=4w^3-g_2w-g_3</math>
 +
 +
 +
 +
 +
 +
==스텀-리우빌==
 +
 +
* [[스텀-리우빌 이론]] 항목에서 자세히 다룸
 +
 +
[[스텀-리우빌 이론|스텀-리우빌 이론]]
 +
 +
 +
 +
 +
 +
==재미있는 사실==
 +
 +
 +
 +
* 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
 +
 +
 +
 +
 +
 +
==역사==
 +
 +
 +
 +
* http://www.google.com/search?hl=en&tbs=tl:1&q=differential+equation
 +
* [[수학사 연표]]
 +
 +
 +
 +
 +
 +
==메모==
 +
 +
* qualitative study
 +
 +
 +
 +
==== 하위페이지 ====
 +
 +
* [[미분방정식]]
 +
** [[Fuchsian 미분방정식(Fuchsian differential equation)]]
 +
** [[그린 함수(Green's function)]]
 +
** [[리만 미분방정식]]
 +
** [[리카티 미분방정식]]
 +
** [[맴돌이군과 미분방정식]]
 +
** [[베르누이 미분방정식]]
 +
** [[베셀 미분방정식]]
 +
** [[스텀-리우빌 이론]]
 +
** [[오일러 미분방정식]]
 +
** [[완전미분방정식]]
 +
** [[이계 미분방정식]]
 +
*** [[상수계수 이계 선형미분방정식]]
 +
*** [[이계 선형 미분방정식]]
 +
** [[일계 선형미분방정식]]
 +
** [[정규특이점(regular singular points)]]
 +
** [[치환적분과 변수분리형 미분방정식]]
 +
** [[팽르베 미분방정식(Painlevé Equations)]]
 +
** [[호인 미분방정식(Heun's equation)]]
 +
 +
 +
 +
 +
 +
==관련된 항목들==
 +
 +
* [[상미분방정식]]
 +
* [[편미분방정식]]
 +
* [[초기하 미분방정식(Hypergeometric differential equations)]]
 +
* [[직교다항식과 special functions|Special functions]]
 +
* [[불가능성의 정리들]]
 +
 +
 +
 +
 +
 +
==수학용어번역==
 +
 +
* http://www.google.com/dictionary?langpair=en|ko&q=
 +
* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]
 +
** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 +
* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 +
 +
 +
 +
 +
 +
==사전 형태의 자료==
 +
 +
* [http://ko.wikipedia.org/wiki/%EC%83%81%EB%AF%B8%EB%B6%84_%EB%B0%A9%EC%A0%95%EC%8B%9D http://ko.wikipedia.org/wiki/상미분_방정식]
 +
* http://ko.wikipedia.org/wiki/
 +
* http://en.wikipedia.org/wiki/differential_equation
 +
* http://www.wolframalpha.com/input/?i=
 +
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 +
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 +
** http://www.research.att.com/~njas/sequences/?q=
 +
 +
 +
==리뷰, 에세이, 강의노트==
 +
* [http://www.jstor.org/stable/2687502 What It Means to Understand a Differential Equation]
 +
** John H. Hubbard, <cite>The College Mathematics Journal</cite>, Vol. 25, No. 5 (Nov., 1994), pp. 372-384
 +
* [http://www.jstor.org/stable/2320204 Elementary Quadratures of Ordinary Differential Equations]
 +
** Li Hong-Xiang, <cite>[http://www.jstor.org/action/showPublication?journalCode=amermathmont The American Mathematical Monthly]</cite>, Vol. 89, No. 3 (Mar., 1982), pp. 198-208
 +
* [http://www.jstor.org/stable/3617402 Symmetry and Differential Equations]
 +
** J. V. Greenman, <cite>[http://www.jstor.org/action/showPublication?journalCode=mathgaze The Mathematical Gazette]</cite>, Vol. 61, No. 418 (Dec., 1977), pp. 279-283
 +
* [http://www.jstor.org/stable/2318252 Anatomy of the Ordinary Differential Equation]
 +
** W. T. Reid, <cite>[http://www.jstor.org/action/showPublication?journalCode=amermathmont The American Mathematical Monthly]</cite>, Vol. 82, No. 10 (Dec., 1975), pp. 971-984
 +
*
 +
** T. Craig
 +
 +
 +
 +
 +
 +
 +
 +
 +
 +
==관련링크와 웹페이지==
 +
 +
* http://eqworld.ipmnet.ru/en/solutions/ode.htm
 +
 +
 +
 +
[[분류:미분방정식]]
 +
 +
== 노트 ==
 +
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q11214 Q11214]
 +
===말뭉치===
 +
# The Differential Equation says it well, but is hard to use.<ref name="ref_7488ce2d">[https://www.mathsisfun.com/calculus/differential-equations.html Differential Equations]</ref>
 +
# Creating a differential equation is the first major step.<ref name="ref_7488ce2d" />
 +
# In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives.<ref name="ref_39f5d84a">[https://byjus.com/maths/differential-equation/ Differential Equations (Definition, Types, Order, Degree, Examples)]</ref>
 +
# The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions.<ref name="ref_39f5d84a" />
 +
# One of the easiest ways to solve the differential equation is by using explicit formulas.<ref name="ref_39f5d84a" />
 +
# A differential equation contains derivatives which are either partial derivatives or ordinary derivatives.<ref name="ref_39f5d84a" />
 +
# 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.<ref name="ref_9e5ce038">[https://en.wikipedia.org/wiki/Differential_equation Differential equation]</ref>
 +
# A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.<ref name="ref_9e5ce038" />
 +
# Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation.<ref name="ref_9e5ce038" />
 +
# A differential equation is an equation involving a function and its derivatives.<ref name="ref_746478e9">[https://www.wolframalpha.com/examples/mathematics/differential-equations/ Alpha Examples: Differential Equations]</ref>
 +
# It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on whether or not partial derivatives are involved.<ref name="ref_746478e9" />
 +
# The first definition that we should cover should be that of differential equation.<ref name="ref_4a3abd97">[https://tutorial.math.lamar.edu/classes/de/definitions.aspx Differential Equations]</ref>
 +
# There is one differential equation that everybody probably knows, that is Newton’s Second Law of Motion.<ref name="ref_4a3abd97" />
 +
# A differential equation is called an ordinary differential equation, abbreviated by ode, if it has ordinary derivatives in it.<ref name="ref_4a3abd97" />
 +
# Likewise, a differential equation is called a partial differential equation, abbreviated by pde, if it has partial derivatives in it.<ref name="ref_4a3abd97" />
 +
# We also investigate how direction fields can be used to determine some information about the solution to a differential equation without actually having the solution.<ref name="ref_8e70e4aa">[https://tutorial.math.lamar.edu/classes/de/de.aspx Differential Equations]</ref>
 +
# Series Solutions – In this section we define ordinary and singular points for a differential equation.<ref name="ref_8e70e4aa" />
 +
# We also show who to construct a series solution for a differential equation about an ordinary point.<ref name="ref_8e70e4aa" />
 +
# rd order differential equation just to say that we looked at one with order higher than 2nd.<ref name="ref_8e70e4aa" />
 +
# A differential equation is an equation involving derivatives.<ref name="ref_bd4e0810">[https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter26/section01.html 26.1 Introduction to Differential Equations]</ref>
 +
# You can find the general solution to any separable first order differential equation by integration, (or as it is sometimes referred to, by "quadrature").<ref name="ref_bd4e0810" />
 +
# Suppose we have a first order differential equation that is not separable, so we cannot reduce its solution to quadratures directly.<ref name="ref_bd4e0810" />
 +
# at least look at what a differential equation actually is.<ref name="ref_5b8ac86e">[https://www.khanacademy.org/math/ap-calculus-ab/ab-differential-equations-new/ab-7-1/v/differential-equation-introduction Differential equations introduction (video)]</ref>
 +
# And you might have just caught from how I described it that the solution to a differential equation is a function, or a class of functions.<ref name="ref_5b8ac86e" />
 +
# We'll verify that these indeed are solutions for I guess this is really just one differential equation represented in different ways.<ref name="ref_5b8ac86e" />
 +
# But you'll hopefully appreciate what a solution to a differential equation looks like.<ref name="ref_5b8ac86e" />
 +
# Differential equation, mathematical statement containing one or more derivatives—that is, terms representing the rates of change of continuously varying quantities.<ref name="ref_9bb2783d">[https://www.britannica.com/science/differential-equation Differential equation]</ref>
 +
# When the function involved in the equation depends on only a single variable, its derivatives are ordinary derivatives and the differential equation is classed as an ordinary differential equation.<ref name="ref_9bb2783d" />
 +
# On the other hand, if the function depends on several independent variables, so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation.<ref name="ref_9bb2783d" />
 +
# Whichever the type may be, a differential equation is said to be of the nth order if it involves a derivative of the nth order but no derivative of an order higher than this.<ref name="ref_9bb2783d" />
 +
# Learning Objectives Calculate the order and degree of a differential equation.<ref name="ref_5f74da62">[https://courses.lumenlearning.com/boundless-calculus/chapter/differential-equations/ Differential Equations]</ref>
 +
# The higher the order of the differential equation, the more arbitrary constants need to be added to the general solution.<ref name="ref_5f74da62" />
 +
# The “order” of a differential equation depends on the derivative of the highest order in the equation.<ref name="ref_5f74da62" />
 +
# The “degree” of a differential equation, similarly, is determined by the highest exponent on any variables involved.<ref name="ref_5f74da62" />
 +
# Also as we have seen so far, a differential equation typically has an infinite number of solutions.<ref name="ref_f6a04ca7">[https://www.whitman.edu/mathematics/calculus_online/section17.01.html 17.1 First Order Differential Equations]</ref>
 +
# Solve a differential equation analytically by using the dsolve function, with or without initial conditions.<ref name="ref_c2ee69ed">[https://www.mathworks.com/help/symbolic/solve-a-single-differential-equation.html Solve Differential Equation]</ref>
 +
# First-Order Linear ODE Solve this differential equation.<ref name="ref_c2ee69ed" />
 +
# Solve this third-order differential equation with three initial conditions.<ref name="ref_c2ee69ed" />
 +
# The last example is the Airy differential equation, whose solution is called the Airy function.<ref name="ref_c2ee69ed" />
 +
# Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function \(y=f(x)\) and its derivative, known as a differential equation.<ref name="ref_d94ff5b0">[https://math.libretexts.org/Courses/Monroe_Community_College/MTH_211_Calculus_II/Chapter_8%3A_Introduction_to_Differential_Equations/8.1%3A_Basics_of_Differential_Equations 8.1: Basics of Differential Equations]</ref>
 +
# Combining like terms leads to the expression \(6x+11\), which is equal to the right-hand side of the differential equation.<ref name="ref_d94ff5b0" />
 +
# A differential equation is an equation involving an unknown function \(y=f(x)\) and one or more of its derivatives.<ref name="ref_d94ff5b0" />
 +
# Consider the equation \(y′=3x^2,\) which is an example of a differential equation because it includes a derivative.<ref name="ref_d94ff5b0" />
 +
# A differential equation is an equation involving terms that are derivatives (or differentials).<ref name="ref_0018c748">[https://ncatlab.org/nlab/show/differential+equation differential equation in nLab]</ref>
 +
# A partial differential equation need not have any solution at all.<ref name="ref_6edac5b5">[https://encyclopediaofmath.org/wiki/Differential_equation,_partial Differential equation, partial]</ref>
 +
# If the partial differential equation being considered is the Euler equation for a problem of variational calculus in more dimensions, a variational method is often employed.<ref name="ref_6edac5b5" />
 +
# The order of a partial differential equation is the order of the highest derivative involved.<ref name="ref_b3c4e8be">[http://www.scholarpedia.org/article/Partial_differential_equation Partial differential equation]</ref>
 +
# A solution (or a particular solution) to a partial differential equation is a function that solves the equation or, in other words, turns it into an identity when substituted into the equation.<ref name="ref_b3c4e8be" />
 +
# A differential equation can look pretty intimidating, with lots of fancy math symbols.<ref name="ref_50027daa">[https://www.myphysicslab.com/explain/what-is-a-diff-eq-en.html myPhysicsLab What Is A Differential Equation?]</ref>
 +
# Each of those variables has a differential equation saying how that variable evolves over time.<ref name="ref_50027daa" />
 +
# The task is to find a function whose various derivatives fit the differential equation over a long span of time.<ref name="ref_50027daa" />
 +
# It is easy to confirm that you have a solution: just plug the solution in to the differential equation!<ref name="ref_50027daa" />
 +
# The final few pages of this class will be devoted to an introduction to differential equation.<ref name="ref_b53ac4e5">[https://ltcconline.net/greenl/courses/117/DiffEQ/diffEQDefs.htm Differential Equations]</ref>
 +
# A differential equation is an equation (you will see an " = " sign) that has derivatives.<ref name="ref_b53ac4e5" />
 +
# If y = f(x) is a solution to a differential equation, then if we plug " y " into the equation, we get a true statement.<ref name="ref_b53ac4e5" />
 +
# We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method.<ref name="ref_85bea581">[https://jmlr.org/papers/v17/15-084.html A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights]</ref>
 +
# The aim of this paper is to study a boundary value problem of the hybrid differential equation with linear and nonlinear perturbations.<ref name="ref_baf4902d">[https://www.hindawi.com/journals/ijde/ International Journal of Differential Equations]</ref>
 +
# The example of a cooling coffee cup is used to find the differential equation and solve it using differentiation.<ref name="ref_2f5e7f28">[https://www.tudelft.nl/en/eemcs/study/online-education/math-explained/ordinary-differential-equations/ Ordinary Differential Equations]</ref>
 +
# If we solve a first order differential equation by variables separable method, we necessarily have to introduce an arbitrary constant as soon as the integration is performed.<ref name="ref_4f329b28">[https://www.toppr.com/guides/maths/differential-equations/general-and-particular-solutions-of-a-differential-equation/ General and Particular Differential Equations Solutions: Videos, Examples]</ref>
 +
# Similarly, the general solution of a second order differential equation will contain 2 necessary arbitrary constants and so on.<ref name="ref_4f329b28" />
 +
# A Particular Solution of a differential equation is a solution obtained from the General Solution by assigning specific values to the arbitrary constants.<ref name="ref_4f329b28" />
 +
# The function f(t) must satisfy the differential equation in order to be a solution.<ref name="ref_4f329b28" />
 +
# In this tutorial we will show you how to define an ordinary differential equation (ODE) in the Fitting function Builder dialog and perform a fit of the data using this fitting function.<ref name="ref_a84801df">[https://www.originlab.com/doc/Tutorials/Fitting-Ordinary-Differential-Equation Fitting with an Ordinary Differential Equation]</ref>
 +
# In this tutorial, we will use a first order ordinary differential equation as an example: where a is a parameter in the ordinary differential equation and y0 is the initial value for the ODE.<ref name="ref_a84801df" />
 +
===소스===
 +
<references />
 +
 +
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q11214 Q11214]
 +
===Spacy 패턴 목록===
 +
* [{'LOWER': 'differential'}, {'LEMMA': 'equation'}]
 +
* [{'LOWER': 'differential'}, {'LOWER': 'equation'}, {'OP': '*'}, {'LOWER': 'elementary'}, {'LOWER': 'mathematics'}, {'LEMMA': ')'}]

2021년 2월 17일 (수) 05:43 기준 최신판

개요

  • 미분방정식은 자연현상을 기술하는 수학적인 언어
  • 함수를 계수로 하여 미지수가 되는 일변수 함수와 고계도함수 사이에 만족되는 방정식을 말함
  • 학부과정에서는 상미분방정식 과목과 편미분방정식이 있음
  • 미분방정식의 해를 적당한 클래스의 함수(가령 초등함수, 초등함수의 적분) 들을 이용하여 표현하는 문제(solvability, integrability, quadrature)
  • 분류법
    • 미분방정식의 계(order)
    • 선형미분방정식과 비선형미분방정식
    • 상미분방정식과 편미분방정식



일계 미분방정식



이계 선형미분방정식

  • 호인 미분방정식(Heun's equation)\[\frac {d^2w}{dz^2} + \left[\frac{\gamma}{z}+ \frac{\delta}{z-1} + \frac{\epsilon}{z-d} \right] \frac {dw}{dz} + \frac {\alpha \beta z -q} {z(z-1)(z-d)} w = 0\] (여기서 \(\epsilon=\alpha+\beta-\gamma-\delta+1\))



비선형 미분방저식



스텀-리우빌

스텀-리우빌 이론



재미있는 사실



역사



메모

  • qualitative study


하위페이지



관련된 항목들



수학용어번역



사전 형태의 자료


리뷰, 에세이, 강의노트





관련링크와 웹페이지

노트

위키데이터

말뭉치

  1. The Differential Equation says it well, but is hard to use.[1]
  2. Creating a differential equation is the first major step.[1]
  3. In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives.[2]
  4. The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions.[2]
  5. One of the easiest ways to solve the differential equation is by using explicit formulas.[2]
  6. A differential equation contains derivatives which are either partial derivatives or ordinary derivatives.[2]
  7. 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.[3]
  8. A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.[3]
  9. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation.[3]
  10. A differential equation is an equation involving a function and its derivatives.[4]
  11. It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on whether or not partial derivatives are involved.[4]
  12. The first definition that we should cover should be that of differential equation.[5]
  13. There is one differential equation that everybody probably knows, that is Newton’s Second Law of Motion.[5]
  14. A differential equation is called an ordinary differential equation, abbreviated by ode, if it has ordinary derivatives in it.[5]
  15. Likewise, a differential equation is called a partial differential equation, abbreviated by pde, if it has partial derivatives in it.[5]
  16. We also investigate how direction fields can be used to determine some information about the solution to a differential equation without actually having the solution.[6]
  17. Series Solutions – In this section we define ordinary and singular points for a differential equation.[6]
  18. We also show who to construct a series solution for a differential equation about an ordinary point.[6]
  19. rd order differential equation just to say that we looked at one with order higher than 2nd.[6]
  20. A differential equation is an equation involving derivatives.[7]
  21. You can find the general solution to any separable first order differential equation by integration, (or as it is sometimes referred to, by "quadrature").[7]
  22. Suppose we have a first order differential equation that is not separable, so we cannot reduce its solution to quadratures directly.[7]
  23. at least look at what a differential equation actually is.[8]
  24. And you might have just caught from how I described it that the solution to a differential equation is a function, or a class of functions.[8]
  25. We'll verify that these indeed are solutions for I guess this is really just one differential equation represented in different ways.[8]
  26. But you'll hopefully appreciate what a solution to a differential equation looks like.[8]
  27. Differential equation, mathematical statement containing one or more derivatives—that is, terms representing the rates of change of continuously varying quantities.[9]
  28. When the function involved in the equation depends on only a single variable, its derivatives are ordinary derivatives and the differential equation is classed as an ordinary differential equation.[9]
  29. On the other hand, if the function depends on several independent variables, so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation.[9]
  30. Whichever the type may be, a differential equation is said to be of the nth order if it involves a derivative of the nth order but no derivative of an order higher than this.[9]
  31. Learning Objectives Calculate the order and degree of a differential equation.[10]
  32. The higher the order of the differential equation, the more arbitrary constants need to be added to the general solution.[10]
  33. The “order” of a differential equation depends on the derivative of the highest order in the equation.[10]
  34. The “degree” of a differential equation, similarly, is determined by the highest exponent on any variables involved.[10]
  35. Also as we have seen so far, a differential equation typically has an infinite number of solutions.[11]
  36. Solve a differential equation analytically by using the dsolve function, with or without initial conditions.[12]
  37. First-Order Linear ODE Solve this differential equation.[12]
  38. Solve this third-order differential equation with three initial conditions.[12]
  39. The last example is the Airy differential equation, whose solution is called the Airy function.[12]
  40. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function \(y=f(x)\) and its derivative, known as a differential equation.[13]
  41. Combining like terms leads to the expression \(6x+11\), which is equal to the right-hand side of the differential equation.[13]
  42. A differential equation is an equation involving an unknown function \(y=f(x)\) and one or more of its derivatives.[13]
  43. Consider the equation \(y′=3x^2,\) which is an example of a differential equation because it includes a derivative.[13]
  44. A differential equation is an equation involving terms that are derivatives (or differentials).[14]
  45. A partial differential equation need not have any solution at all.[15]
  46. If the partial differential equation being considered is the Euler equation for a problem of variational calculus in more dimensions, a variational method is often employed.[15]
  47. The order of a partial differential equation is the order of the highest derivative involved.[16]
  48. A solution (or a particular solution) to a partial differential equation is a function that solves the equation or, in other words, turns it into an identity when substituted into the equation.[16]
  49. A differential equation can look pretty intimidating, with lots of fancy math symbols.[17]
  50. Each of those variables has a differential equation saying how that variable evolves over time.[17]
  51. The task is to find a function whose various derivatives fit the differential equation over a long span of time.[17]
  52. It is easy to confirm that you have a solution: just plug the solution in to the differential equation![17]
  53. The final few pages of this class will be devoted to an introduction to differential equation.[18]
  54. A differential equation is an equation (you will see an " = " sign) that has derivatives.[18]
  55. If y = f(x) is a solution to a differential equation, then if we plug " y " into the equation, we get a true statement.[18]
  56. We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method.[19]
  57. The aim of this paper is to study a boundary value problem of the hybrid differential equation with linear and nonlinear perturbations.[20]
  58. The example of a cooling coffee cup is used to find the differential equation and solve it using differentiation.[21]
  59. If we solve a first order differential equation by variables separable method, we necessarily have to introduce an arbitrary constant as soon as the integration is performed.[22]
  60. Similarly, the general solution of a second order differential equation will contain 2 necessary arbitrary constants and so on.[22]
  61. A Particular Solution of a differential equation is a solution obtained from the General Solution by assigning specific values to the arbitrary constants.[22]
  62. The function f(t) must satisfy the differential equation in order to be a solution.[22]
  63. In this tutorial we will show you how to define an ordinary differential equation (ODE) in the Fitting function Builder dialog and perform a fit of the data using this fitting function.[23]
  64. In this tutorial, we will use a first order ordinary differential equation as an example: where a is a parameter in the ordinary differential equation and y0 is the initial value for the ODE.[23]

소스

  1. ↑ 1.0 1.1 Differential Equations
  2. ↑ 2.0 2.1 2.2 2.3 Differential Equations (Definition, Types, Order, Degree, Examples)
  3. ↑ 3.0 3.1 3.2 Differential equation
  4. ↑ 4.0 4.1 Alpha Examples: Differential Equations
  5. ↑ 5.0 5.1 5.2 5.3 Differential Equations
  6. ↑ 6.0 6.1 6.2 6.3 Differential Equations
  7. ↑ 7.0 7.1 7.2 26.1 Introduction to Differential Equations
  8. ↑ 8.0 8.1 8.2 8.3 Differential equations introduction (video)
  9. ↑ 9.0 9.1 9.2 9.3 Differential equation
  10. ↑ 10.0 10.1 10.2 10.3 Differential Equations
  11. ↑ 17.1 First Order Differential Equations
  12. ↑ 12.0 12.1 12.2 12.3 Solve Differential Equation
  13. ↑ 13.0 13.1 13.2 13.3 8.1: Basics of Differential Equations
  14. ↑ differential equation in nLab
  15. ↑ 15.0 15.1 Differential equation, partial
  16. ↑ 16.0 16.1 Partial differential equation
  17. ↑ 17.0 17.1 17.2 17.3 myPhysicsLab What Is A Differential Equation?
  18. ↑ 18.0 18.1 18.2 Differential Equations
  19. ↑ A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights
  20. ↑ International Journal of Differential Equations
  21. ↑ Ordinary Differential Equations
  22. ↑ 22.0 22.1 22.2 22.3 General and Particular Differential Equations Solutions: Videos, Examples
  23. ↑ 23.0 23.1 Fitting with an Ordinary Differential Equation

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'differential'}, {'LEMMA': 'equation'}]
  • [{'LOWER': 'differential'}, {'LOWER': 'equation'}, {'OP': '*'}, {'LOWER': 'elementary'}, {'LOWER': 'mathematics'}, {'LEMMA': ')'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=미분방정식&oldid=52296"