Differential equations · 微分方程
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| decay/dɪˈkeɪ/ | 衰减 | shuāi jiǎn |
| differential equation/ˌdɪfəˈrenʃl ɪˈkweɪʒn/ | 微分方程 | wēi fēn fāng chéng |
| rate of change/reɪt ɒv tʃeɪndʒ/ | 变化率 | biàn huà lǜ |
| initial condition/ɪˈnɪʃl kənˈdɪʃn/ | 初始条件 | chū shǐ tiáo jiàn |
| separable/ˈsepərəbl/ | 可分离变量 | kě fēn lí biàn liàng |
| constant of integration/ˈkɒnstənt ɒv ˌɪntɪˈɡreɪʃn/ | 积分常数 | jī fēn cháng shù |
| particular solution/pəˈtɪkjʊlə səˈluːʃn/ | 特解 | tè jiě |
| general solution/ˈdʒenərəl səˈluːʃn/ | 通解 | tōng jiě |
The equation that predicts the future
- How fast does a population grow? How quickly does a radioactive substance decay 衰减? How does a cup of coffee cool?
- All these questions lead to differential equations 微分方程 — equations that link a quantity to its rate of change 变化率. Solving them predicts the future.
预测未来的方程
- 一个种群增长多快?一种放射性物质衰变多快?一杯咖啡如何冷却?
- 所有这些问题都引向微分方程(differential equations)——把一个量和它的变化率联系起来的方程。求解它们就预测未来。
What is a differential equation?
- A differential equation links a quantity to its rate of change.
- Example: $\dfrac{dy}{dx} = ky$ says the rate of growth is proportional to the current value — exponential growth.
Real-world example. Newton's law of cooling: $\dfrac{dT}{dt} = -k(T - T_{\text{room}})$. The rate of cooling is proportional to how much hotter the object is than its surroundings.
The constant gives a family of curves; an initial condition 初始条件 picks out one
什么是微分方程?
- 一个微分方程把一个量和它的变化率联系起来。
- 例子:$\dfrac{dy}{dx} = ky$ 说增长率与当前值成正比——指数增长。
现实世界的例子。 牛顿冷却定律:$\dfrac{dT}{dt} = -k(T - T_{\text{room}})$。冷却率与物体比它的环境热多少成正比。
常数给出一族曲线;一个初始条件挑出一条
The slope field · 斜率场
dy/dx = a y
A differential equation gives the slope · 斜率 everywhere — the solution is the curve that follows those slopes from y₀. · 一个微分方程给出到处的斜率——解是从 y₀ 跟随那些斜率的曲线。
A differential equation links a quantity to its rate of change. · 一个微分方程把一个量和它的变化率联系起来。
That is the defining feature: the equation involves both y and dy/dx (or higher derivatives). · 那就是定义性的特征:方程同时涉及 y 和 dy/dx(或更高阶的导数)。
Separable 可分离变量 equations
- For a separable first-order equation: put all $y$ terms on one side, all $x$ terms on the other, then integrate both sides.
- Example: $\dfrac{dy}{dx} = xy \Rightarrow \dfrac{1}{y}\,dy = x\,dx \Rightarrow \ln|y| = \dfrac{x^2}{2} + C$.
Don't forget the constant of integration 积分常数. When you integrate both sides, add $+C$ to one side. This constant determines which particular solution 特解 you get.
可分离的方程
- 对一个可分离的(separable)一阶方程:把所有 $y$ 项放在一边,所有 $x$ 项放在另一边,然后对两边积分。
- 示例：$\dfrac{dy}{dx} = xy \Rightarrow \dfrac{1}{y}\,dy = x\,dx \Rightarrow \ln|y| = \dfrac{x^2}{2} + C$。
不要忘记积分常数。 当你对两边积分时,在一边加上 $+C$。这个常数决定你得到哪个特解。
To solve a separable first-order differential equation, you: · 要解一个可分离的一阶微分方程,你:
Separate the variables, then integrate each side. · 分离变量,然后对每一边积分。
General vs particular solutions
- The general solution 通解 contains a constant (a whole family of curves).
- An initial condition (a known value) fixes the constant → the particular solution.
通解对特解
- 通解(general solution)包含一个常数(一整族曲线)。
- 一个初始条件(initial condition,一个已知的值)固定这个常数 → 特解(particular solution)。
An initial condition is used to: · 一个初始条件用来:
A known value fixes the arbitrary constant, turning the general solution into the particular one. · 一个已知的值固定任意常数,把通解变成特解。
The general solution of a differential equation contains an arbitrary constant. · 一个微分方程的通解包含一个任意常数。
Integrating introduces a constant, so the general solution is a whole family of curves. · 积分引入一个常数,所以通解是一整族曲线。
Worked example
- Solve $\dfrac{dy}{dx} = xy$ with $y = 1$ at $x = 0$.
- Separate: $\dfrac{1}{y}\,dy = x\,dx$. Integrate: $\ln|y| = \dfrac{x^2}{2} + C$.
- Apply $y(0) = 1$: $\ln 1 = 0 + C \Rightarrow C = 0$.
- So $\ln y = \dfrac{x^2}{2} \Rightarrow y = e^{x^2/2}$.
allow_no_figure: Differential equations are solved algebraically — the solution process (separation of variables, integration) is the key content, not a diagram.
- Solve a first order differential equation to model real-life rates of change.
示例
- 解决 $\dfrac{dy}{dx} = xy$ 与 $y = 1$ 在 $x = 0$.
- 分离:$\dfrac{1}{y}\,dy = x\,dx$。积分:$\ln|y| = \dfrac{x^2}{2} + C$。
- 应用 $y(0) = 1$:$\ln 1 = 0 + C \Rightarrow C = 0$。
- 所以 $\ln y = \dfrac{x^2}{2} \Rightarrow y = e^{x^2/2}$。
allow_no_figure: 微分方程通过代数方法求解——解题过程（变量分离、积分）是关键内容，而非图表。
- 解一阶微分方程(first order differential equation)来建模现实(real-life)中的变化率。
Solve dy/dx = xy with y(0) = 1. At x = 2, y = e^(x²/2). Find y (2 dp). · 解 dy/dx = xy,y(0) = 1。在 x = 2,y = e^(x²/2)。求 y(2 位小数)。
y = e^(x²/2). At x = 2: y = e² ≈ 7.39. · y = e^(x²/2)。在 x = 2:y = e² ≈ 7.39。
If ln y = x²/2 + C and y(0) = 1, what is C? · 如果 ln y = x²/2 + C 且 y(0) = 1,C 是多少?
ln 1 = 0 + C, so C = 0. · ln 1 = 0 + C,所以 C = 0。
You've got it
- solve a separable equation by splitting variables and integrating both sides
- the general solution has a constant; an initial condition gives the particular solution
- a differential equation links a quantity to its rate of change
你掌握了
- 通过分离变量并对两边积分来解一个可分离的方程
- 通解有一个常数;一个初始条件给出特解
- 一个微分方程把一个量和它的变化率联系起来