미분방정식

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

• 일계선형미분방정식\[\frac{dy}{dt}+a(t)y=b(t)\]
• 완전미분방정식\[M_y=N_x\]를 만족시키는 \(M(x, y)\, dx + N(x, y)\, dy = 0\) 꼴의 미분방정식
• 다음 미분방정식들은 비선형이다
• 리카티 미분방정식\[y' = A(x)+ B(x)y + C(x)y^2, A(x)\neq 0, C(x)\neq 0\]
• 베르누이 미분방정식\[y'+ P(x)y = Q(x)y^n\]

이계 선형미분방정식

• 다음 형태로 주어지는 미분방정식을 이계선형미분방정식이라 함\[\frac{d^2y}{dx^2}+p(x)\frac{dy}{dx}+q(x)y=g(x)\]
• 상수계수 이계 선형미분방정식\[ay''+by'+cy=0\]
• Airy 미분방정식\[y'' - xy = 0\]
• 베셀 미분방정식\[x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0\]
• 에르미트 다항식(Hermite polynomials)\[y''-2xy'+\lambda y=0\]
• 르장드르 다항식\[(1-x^2)y''-2xy'+\lambda(\lambda+1) y=0\]
• 체비셰프 다항식\[(1-x^2)y''-xy'+\lambda^2 y=0\]
• 라게르 미분방정식\[xy''+(1-x)y'+\lambda y=0\]
• 오일러 미분방정식\[x^2\frac{d^2y}{dx^2}+\alpha x\frac{dy}{dx}+\beta y=0\]
• 초기하 미분방정식(Hypergeometric differential equations)\[z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0\]
• 리만 미분방정식\[\frac{d^2w}{dz^2} + \left[ \frac{1-\alpha-\alpha'}{z-a} + \frac{1-\beta-\beta'}{z-b} + \frac{1-\gamma-\gamma'}{z-c} \right] \frac{dw}{dz}+\left[ \frac{\alpha\alpha' (a-b)(a-c)} {z-a} +\frac{\beta\beta' (b-c)(b-a)} {z-b} +\frac{\gamma\gamma' (c-a)(c-b)} {z-c} \right] \frac{w}{(z-a)(z-b)(z-c)}=0\] 여기서 \(\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1\)

• 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

• 팽르베 미분방정식
• 바이어슈트라스의 타원함수\[(\frac{dw}{dz})^2=4w^3-g_2w-g_3\]

스텀-리우빌

• 스텀-리우빌 이론 항목에서 자세히 다룸

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)
• Special functions
• 불가능성의 정리들

수학용어번역

• http://www.google.com/dictionary?langpair=en%7Cko&q=
• 대한수학회 수학 학술 용어집
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 대한수학회 수학용어한글화 게시판

사전 형태의 자료

• 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=
• NIST Digital Library of Mathematical Functions
• The On-Line Encyclopedia of Integer Sequences
  • http://www.research.att.com/~njas/sequences/?q=

리뷰, 에세이, 강의노트

• What It Means to Understand a Differential Equation
  • John H. Hubbard, The College Mathematics Journal, Vol. 25, No. 5 (Nov., 1994), pp. 372-384
• Elementary Quadratures of Ordinary Differential Equations
  • Li Hong-Xiang, The American Mathematical Monthly, Vol. 89, No. 3 (Mar., 1982), pp. 198-208
• Symmetry and Differential Equations
  • J. V. Greenman, The Mathematical Gazette, Vol. 61, No. 418 (Dec., 1977), pp. 279-283
• Anatomy of the Ordinary Differential Equation
  • W. T. Reid, The American Mathematical Monthly, Vol. 82, No. 10 (Dec., 1975), pp. 971-984
• • T. Craig

관련링크와 웹페이지

• http://eqworld.ipmnet.ru/en/solutions/ode.htm

노트

위키데이터

• ID : Q11214

말뭉치

1. The Differential Equation says it well, but is hard to use.
2. Creating a differential equation is the first major step.
3. In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives.
4. The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions.
5. One of the easiest ways to solve the differential equation is by using explicit formulas.
6. A differential equation contains derivatives which are either partial derivatives or ordinary derivatives.
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.
8. A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.
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.
10. A differential equation is an equation involving a function and its derivatives.
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.
12. The first definition that we should cover should be that of differential equation.
13. There is one differential equation that everybody probably knows, that is Newton’s Second Law of Motion.
14. A differential equation is called an ordinary differential equation, abbreviated by ode, if it has ordinary derivatives in it.
15. Likewise, a differential equation is called a partial differential equation, abbreviated by pde, if it has partial derivatives in it.
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.
17. Series Solutions – In this section we define ordinary and singular points for a differential equation.
18. We also show who to construct a series solution for a differential equation about an ordinary point.
19. rd order differential equation just to say that we looked at one with order higher than 2nd.
20. A differential equation is an equation involving derivatives.
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").
22. Suppose we have a first order differential equation that is not separable, so we cannot reduce its solution to quadratures directly.
23. at least look at what a differential equation actually is.
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.
25. We'll verify that these indeed are solutions for I guess this is really just one differential equation represented in different ways.
26. But you'll hopefully appreciate what a solution to a differential equation looks like.
27. Differential equation, mathematical statement containing one or more derivatives—that is, terms representing the rates of change of continuously varying quantities.
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.
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.
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.
31. Learning Objectives Calculate the order and degree of a differential equation.
32. The higher the order of the differential equation, the more arbitrary constants need to be added to the general solution.
33. The “order” of a differential equation depends on the derivative of the highest order in the equation.
34. The “degree” of a differential equation, similarly, is determined by the highest exponent on any variables involved.
35. Also as we have seen so far, a differential equation typically has an infinite number of solutions.
36. Solve a differential equation analytically by using the dsolve function, with or without initial conditions.
37. First-Order Linear ODE Solve this differential equation.
38. Solve this third-order differential equation with three initial conditions.
39. The last example is the Airy differential equation, whose solution is called the Airy function.
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.
41. Combining like terms leads to the expression \(6x+11\), which is equal to the right-hand side of the differential equation.
42. A differential equation is an equation involving an unknown function \(y=f(x)\) and one or more of its derivatives.
43. Consider the equation \(y′=3x^2,\) which is an example of a differential equation because it includes a derivative.
44. A differential equation is an equation involving terms that are derivatives (or differentials).
45. A partial differential equation need not have any solution at all.
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.
47. The order of a partial differential equation is the order of the highest derivative involved.
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.
49. A differential equation can look pretty intimidating, with lots of fancy math symbols.
50. Each of those variables has a differential equation saying how that variable evolves over time.
51. The task is to find a function whose various derivatives fit the differential equation over a long span of time.
52. It is easy to confirm that you have a solution: just plug the solution in to the differential equation!
53. The final few pages of this class will be devoted to an introduction to differential equation.
54. A differential equation is an equation (you will see an " = " sign) that has derivatives.
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.
56. We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method.
57. The aim of this paper is to study a boundary value problem of the hybrid differential equation with linear and nonlinear perturbations.
58. The example of a cooling coffee cup is used to find the differential equation and solve it using differentiation.
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.
60. Similarly, the general solution of a second order differential equation will contain 2 necessary arbitrary constants and so on.
61. A Particular Solution of a differential equation is a solution obtained from the General Solution by assigning specific values to the arbitrary constants.
62. The function f(t) must satisfy the differential equation in order to be a solution.
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.
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.