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第146行： − 微分方程可分为几种类型。除了描述方程本身的性质之外，微分方程的类型有助于指导解决方案的选择。常用的区别包括方程是否为: 常微分/偏微分方程、线性/非线性方程和齐次/非齐次方程。这份清单远非详尽无遗; 在特定的情况下，微分方程还有许多非常有用的其他性质和子类。+ 第159行：
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第167行： − 线性微分方程是关于指未知函数及其导数都是线性的微分方程。关于这些方程的理论发展得很好，在多数情况下可以用积分来表示他们的解。+ 第183行：
第183行： − 一般来说，微分方程的解不能用解析解表示，通常会在计算机上利用数值方法求解微分方程。+ 第214行：
第214行： − + 第220行：
第220行： − 线性微分方程经常作为非线性方程的近似形式出现。这些近似只有在受限制的条件下才有效。例如，谐振子方程是非线性摆方程的近似这一情况只有对于小幅度振荡是有效的(见下文)。+ 第239行：
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第297行： − 齐次二阶线性常系数椭圆偏微分方程，也称为拉普拉斯方程+
→Types
Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, and Homogeneous/heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.
Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, and Homogeneous/heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.
微分方程可分为几种类型。除了描述方程本身的性质之外，微分方程的类型有助于指导选择何种微分方程作为解决方案。常见的区别包括方程是否为: 常微分/偏微分方程、线性/非线性方程和齐次/非齐次方程。这份清单远非详尽无遗; 在特定的情况下，微分方程还有许多非常有用的其他性质和子类。
An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable , its derivatives, and some given functions of . The unknown function is generally represented by a variable (often denoted {{mvar|y}}), which, therefore, depends on {{mvar|x}}. Thus {{mvar|x}} is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.
An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable , its derivatives, and some given functions of . The unknown function is generally represented by a variable (often denoted {{mvar|y}}), which, therefore, depends on {{mvar|x}}. Thus {{mvar|x}} is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.
常微分方程中，包含有：只含有一个实变量或复变量的未知函数及其导数以及一些已知的函数。未知函数通常由一个变量(通常由{{mvar|y}}表示)表示，因此这个变量依赖于{{mvar|x}}。因此{{mvar|x}}通常被称为方程式的自变量。“常微分方程 ”一词与偏微分方程一词形成对比，后者可能涉及一个以上的独立变量。
常微分方程中，包含有：只含有一个实变量或复变量的未知函数及其导数以及一些已知的函数。未知函数通常由一个变量(通常由 {{mvar|y}} 表示)表示，因此这个变量依赖于 {{mvar|x}} 。因此 {{mvar|x}} 通常被称为方程式的自变量。“常微分方程”一词与偏微分方程一词相比，后者可能涉及一个以上的独立变量。
Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of integrals.
Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of integrals.
线性微分方程是指方程中未知函数及其导数都是线性的微分方程。关于这些方程的理论发展得很好，在多数情况下可以用积分来表示他们的解。
As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical methods are commonly used for solving differential equations on a computer.
As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical methods are commonly used for solving differential equations on a computer.
一般来说，微分方程的解不能用解析解表示，而通常会在计算机上利用数值方法求解微分方程。
===Partial differential equations===
===Partial differential equations===
A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.
A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.
非线性微分方程是微分方程的一种，但它不是关于未知函数及其导数的线性方程(这里不考虑函数论元中的线性或非线性)。能够精确求解非线性微分方程的方法很少; 那些已有的方法通常依赖于方程具有特定的对称性。非线性微分方程在更长的时间段内表现出非常复杂的行为，具有混沌特性。即使是非线性微分方程解的存在性、唯一性和可扩展性等基本问题，以及非线性偏微分方程初边值问题的适定性问题，也是一个难题（查阅，纳维-斯托克斯方程的存在性和光滑性)。然而，如果微分方程是一个有意义物理过程的正确表述，那么人们期望它有一个解析解。
非线性微分方程是微分方程的一种，但它不是关于未知函数及其导数的线性方程(这里不考虑函数论元中的线性或非线性)。能够精确求解非线性微分方程的方法很少; 那些已有的方法通常依赖于方程具有某种特定的对称性。非线性微分方程在更长的时间段内表现出非常复杂的行为，具有混沌特性。即使是非线性微分方程解的存在性、唯一性和可扩展性等基本问题，以及非线性偏微分方程初边值问题的适定性问题，也是一个难题（查阅，纳维-斯托克斯方程的存在性和光滑性)。然而，如果微分方程是一个有意义物理过程的正确表述，那么人们期望它有一个解析解。
Linear differential equations frequently appear as [[linearization|approximations]] to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).
Linear differential equations frequently appear as [[linearization|approximations]] to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).
Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).
Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).
线性微分方程经常作为非线性方程的近似形式出现。这些近似只有某些限制条件下才有效。例如，谐振子方程是非线性摆方程的近似这一情况只有对于小幅度振荡是有效的(见下文)。
==={{anchor|Second order}} Equation order===
==={{anchor|Second order}} Equation order===
In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones.
In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones.
在第一组例子中，待求解的''u''是''x''的函数，''c''和''ω''是应该已知的常数。常微分方程和偏微分方程的两种广义分类要区分微分方程的线性和非线性，以及区分微分方程的齐次和非齐次。
在第一组例子中，待求解的''u''是''x''的函数，''c''和''ω''是应该已知的常数。常微分方程和偏微分方程这两种广义分类下还要区分微分方程的线性和非线性，以及区分微分方程的齐次和非齐次。
* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the [[Laplace equation]]:
* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the [[Laplace equation]]:
齐次二阶线性常系数椭圆形偏微分方程，也称为拉普拉斯方程