Differential equations
| English | Chinese | Pinyin |
|---|---|---|
| decay | 衰减 | shuāi jiǎn |
| differential equation | 微分方程 | wēi fēn fāng chéng |
| rate of change | 变化率 | biàn huà lǜ |
| initial condition | 初始条件 | chū shǐ tiáo jiàn |
| separable | 可分离变量 | kě fēn lí biàn liàng |
| constant of integration | 积分常数 | jī fēn cháng shù |
| particular solution | 特解 | tè jiě |
| general solution | 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.
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
The slope field
dy/dx = a y
A differential equation gives the slope everywhere — the solution is the curve that follows those slopes from y₀.
A differential equation links a quantity to its rate of change.
That is the defining feature: the equation involves both y and dy/dx (or higher derivatives).
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.
To solve a separable first-order differential equation, you:
Separate the variables, then integrate each side.
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.
An initial condition is used to:
A known value fixes the arbitrary constant, turning the general solution into the particular one.
The general solution of a differential equation contains an arbitrary constant.
Integrating introduces a constant, so the general solution is a whole family of curves.
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.
Solve dy/dx = xy with y(0) = 1. At x = 2, y = e^(x²/2). Find y (2 dp).
y = e^(x²/2). At x = 2: y = e² ≈ 7.39.
If ln y = x²/2 + C and y(0) = 1, what is C?
ln 1 = 0 + C, so 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