Modeling Situations with Differential Equations · 用微分方程建模情境
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| differential equation/ˌdɪfəˈrenʃl ɪˈkweɪʒn/ | 微分方程 | wēi fēn fāng chéng |
Equations that describe change itself
- Many real laws describe not a quantity, but how fast it changes.
- A differential equation 微分方程 relates a function to its derivatives.
- "Population grows at a rate proportional to its size" becomes $\dfrac{dP}{dt}=kP$.
- Solving it recovers the function; this unit is about writing and solving such equations.
描述"变化本身"的方程
- 许多现实规律描述的不是一个量,而是它变化多快。
- 微分方程把一个函数与它的导数联系起来。
- "人口以正比于其规模的速率增长"变成 $\dfrac{dP}{dt}=kP$。
- 解它就还原出那个函数;本单元讲写出并求解这类方程。
A rate equation's slope field · 速率方程的方向场
The equation · 方程 $\tfrac{dy}{dt}=ky$ says the rate is proportional to $y$ — its slope field shows the growth it models. · 方程$\tfrac{dy}{dt}=ky$表示速率与$y$成正比——其方向场展示了所建模的增长模式。
A differential equation relates a function to its... · 微分方程关联了一个函数与其...
It involves the function and its derivative(s). · 它涉及函数及其导数。
Translating words into a rate equation
- The phrase "the rate of change of $y$" is $\dfrac{dy}{dt}$ — the left side of your equation.
- "proportional to $y$" → $ky$; "proportional to the difference from $M$" → $k(M-y)$.
- Match the English description of the rate to an expression in $y$ (and maybe $t$).
- The whole model is: (rate of change) = (that expression).
把文字翻译成速率方程
- 短语"$y$ 的变化率"就是 $\dfrac{dy}{dt}$——你方程的左边。
- "正比于 $y$"→ $ky$;"正比于与 $M$ 的差"→ $k(M-y)$。
- 把关于速率的英文描述匹配成 $y$(也许还有 $t$)的表达式。
- 整个模型就是:(变化率)=(那个表达式)。
"The rate of change of $y$ is proportional to $y$" becomes... · “$y$的变化率与$y$成正比”转化为...
Rate of change $\frac{dy}{dt}$ = $k$ times $y$. · 变化率$\frac{dy}{dt}$ = $k$ × $y$。
Water drains at a rate proportional to the amount $W$. The model is... · 水以与量$W$成正比的速率流出。模型为...
Draining means decreasing → negative rate: $-kW$. · 流出意味着减少 → 负速率：$-kW$。
Common modeling phrases
- Proportional to the amount: $\dfrac{dy}{dt}=ky$ — exponential growth/decay.
- Proportional to the amount present and remaining: $\dfrac{dy}{dt}=ky(M-y)$ — logistic-type.
- Constant rate: $\dfrac{dy}{dt}=c$.
- Reading the phrase tells you the right-hand side.
常见的建模短语
- 正比于现有量: $\dfrac{dy}{dt}=ky$——指数增长/衰减。
- 正比于现有量且剩余量: $\dfrac{dy}{dt}=ky(M-y)$——逻辑斯蒂型。
- 恒定速率: $\dfrac{dy}{dt}=c$。
- 读懂短语就知道右边是什么。
Match each phrase to its rate expression. · 将每个短语与其速率表达式匹配。
Read the phrase to build the right-hand side. · 阅读短语以构建右侧表达式。
A differential equation has a family of solutions
- Unlike an algebraic equation (one unknown number), a differential equation has an unknown function.
- Its solution is usually a family of functions (with a constant), not a single answer.
- A specific member is pinned down later by an initial condition (lesson 7.7).
- For now, the goal is just to write the equation from the situation.
微分方程有一族解
- 与代数方程(一个未知数)不同,微分方程的未知量是一个函数。
- 它的解通常是一族函数(带一个常数),而非单个答案。
- 具体的某一个之后由初始条件确定(7.7 课)。
- 现在,目标只是从情境中写出方程。
$y=kt$ (with no derivative) is a differential equation. · $y=kt$（不含导数）不是微分方程。
A differential equation must involve a derivative. · 微分方程必须包含导数。
The solution of a differential equation is usually a ____ of functions, not a single one. · 微分方程的解通常是函数的____族，而非单个函数。
A constant remains until an initial condition fixes it. · 常数会保留直到初始条件将其确定。
A differential equation involves a derivative — it's a statement about a rate, not the quantity itself. "$y$ is proportional to $t$" ($y=kt$) is not a differential equation; "the rate of change of $y$ is proportional to $y$" ($\frac{dy}{dt}=ky$) is. Look for the "rate of change" language.
微分方程含一个导数——它是关于速率的陈述,而非量本身。"$y$ 正比于 $t$"($y=kt$)不是微分方程;"$y$ 的变化率正比于 $y$"($\frac{dy}{dt}=ky$)才是。寻找"变化率"这样的措辞。
"A tank's water drains at a rate proportional to the amount of water $W$." Write the model.
- Rate of change of $W$: $\dfrac{dW}{dt}$.
- Proportional to $W$, and draining (decreasing) → $\dfrac{dW}{dt}=-kW$ (with $k>0$).
- The negative sign encodes that the water is decreasing.
"一个水箱以正比于水量 $W$ 的速率排水。"写出模型。
- $W$ 的变化率:$\dfrac{dW}{dt}$。
- 正比于 $W$,且在排水(减少)→ $\dfrac{dW}{dt}=-kW$(其中 $k>0$)。
- 负号编码了水在减少。
A differential equation relates a function to its derivatives — it models a rate of change. Translate the words: "rate of change of $y$" is $\frac{dy}{dt}$, "proportional to $y$" is $ky$, and so on. Its solution is a family of functions, pinned down later by an initial condition.
微分方程把一个函数与它的导数联系起来——它建模变化率。翻译文字:"$y$ 的变化率"是 $\frac{dy}{dt}$,"正比于 $y$"是 $ky$,等等。它的解是一族函数,之后由初始条件确定。