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[[분류:미분방정식]]
 
[[분류:미분방정식]]
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== 노트 ==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q11214 Q11214]
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===말뭉치===
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# 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>
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# 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" />
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# 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===
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<references />

2020년 12월 21일 (월) 01:51 판

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
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