# 미분방정식

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## 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

## 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 $$\epsilon=\alpha+\beta-\gamma-\delta+1$$)

## 메모

• qualitative study

## 노트

### 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]

## 메타데이터

### Spacy 패턴 목록

• [{'LOWER': 'differential'}, {'LEMMA': 'equation'}]
• [{'LOWER': 'differential'}, {'LOWER': 'equation'}, {'OP': '*'}, {'LOWER': 'elementary'}, {'LOWER': 'mathematics'}, {'LEMMA': ')'}]