Differential equations (Further Pure 2) · 微分方程(Further Pure 2)
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| differential equation/ˌdɪfəˈrenʃl ɪˈkweɪʒn/ | 微分方程 | wēi fēn fāng chéng |
| integrating factor/ˈɪntɪɡreɪtɪŋ ˈfæktə/ | 积分因子 | jī fēn yīn zi |
| general solution/ˈdʒenərəl səˈluːʃn/ | 通解 | tōng jiě |
| complementary function/ˌkɒmplɪˈmentəri ˈfʌŋkʃn/ | 余函数 | yú hán shù |
| auxiliary equation/ɔːkˈsɪlɪəri ɪˈkweɪʒn/ | 辅助方程 | fǔ zhù fāng chéng |
| boundary condition/ˈbaʊndəri kənˈdɪʃn/ | 边界条件 | biān jiè tiáo jiàn |
| initial condition/ɪˈnɪʃl kənˈdɪʃn/ | 初始条件 | chū shǐ tiáo jiàn |
| particular integral/pəˈtɪkjʊlə ˈɪntɪɡrəl/ | 特积分 | tè jī fēn |
When the unknown is a function
- Ordinary equations solve for a number. A differential equation 微分方程 solves for a whole function — given a rule about its rate of change.
- They model cooling coffee, growing populations, charging capacitors, and swinging pendulums.
当未知量是一个函数
- 普通方程求解一个数。一个微分方程(differential equation)求解一整个函数——给定一条关于它变化率的规则。
- 它们建模冷却的咖啡、增长的种群、充电的电容器和摆动的钟摆。
First-order linear: the integrating factor 积分因子
- For $\dfrac{dy}{dx} + P(x)\,y = Q(x)$, multiply through by the integrating factor:
- This makes the left side collapse to $\dfrac{d}{dx}(\mu y)$, so you can integrate both sides directly.
Worked example. $\dfrac{dy}{dx} + 2y = e^x$. Here $P = 2$, so $\mu = e^{\int 2\,dx} = e^{2x}$. The left side becomes $\dfrac{d}{dx}(e^{2x}y)$.
一阶线性:积分因子
- 对 $\dfrac{dy}{dx} + P(x)\,y = Q(x)$,整个乘以积分因子(integrating factor):
- 这使左边坍缩为 $\dfrac{d}{dx}(\mu y)$,所以你能直接对两边积分。
例题。 $\dfrac{dy}{dx} + 2y = e^x$。这里 $P = 2$,所以 $\mu = e^{\int 2\,dx} = e^{2x}$。左边变成 $\dfrac{d}{dx}(e^{2x}y)$。
Differential equations · 微分方程
dy/dx = a y
The equation sets the slope · 斜率 everywhere; the solution follows those slopes. · 方程处处设定斜率;解沿着那些斜率。
For dy/dx + 2y = eˣ, the integrating factor μ = e^(∫2 dx) is: · 对 dy/dx + 2y = eˣ,积分因子 μ = e^(∫2 dx) 是:
∫2 dx = 2x, so μ = e^(2x). · ∫2 dx = 2x,所以 μ = e^(2x)。
Multiplying by the integrating factor makes the left side d/dx(μy), so you can integrate directly. · 乘以积分因子使左边成为 d/dx(μy),所以你能直接积分。
That is the whole point of the integrating factor — it makes the left side an exact derivative. · 那就是积分因子的全部意义——它使左边成为一个恰当的导数。
Second-order, constant coefficients
- For $a\dfrac{d^2y}{dx^2} + b\dfrac{dy}{dx} + cy = f(x)$, the general solution 通解 has two parts:
- The complementary function 余函数 (CF) comes from the auxiliary equation 辅助方程 $am^2 + bm + c = 0$.
二阶,常系数
- 对 $a\dfrac{d^2y}{dx^2} + b\dfrac{dy}{dx} + cy = f(x)$,通解有两部分:
- 补函数(CF)来自辅助方程(auxiliary equation)$am^2 + bm + c = 0$。
For a constant-coefficient linear equation, the general solution is: · 对一个常系数线性方程,通解是:
General solution = complementary function (RHS = 0) + a particular integral. · 通解 = 补函数(右边 = 0)+ 一个特解。
Reading the auxiliary roots
- The roots of the auxiliary equation decide the shape of the CF:
| roots | complementary function |
|---|---|
| real & distinct $m_1, m_2$ | $Ae^{m_1 x} + Be^{m_2 x}$ |
| repeated $m$ | $(A + Bx)e^{mx}$ |
| complex $p \pm qi$ | $e^{px}(A\cos qx + B\sin qx)$ |
A differential equation has a whole family of solution curves; the boundary conditions 边界条件 pick out one.
读取辅助根
- 辅助方程的根决定 CF 的形状:
| 根 | 补函数 |
|---|---|
| 实且相异 $m_1, m_2$ | $Ae^{m_1 x} + Be^{m_2 x}$ |
| 重根 $m$ | $(A + Bx)e^{mx}$ |
| 复数 $p \pm qi$ | $e^{px}(A\cos qx + B\sin qx)$ |
一个微分方程有一整族解曲线;边界条件挑出一条。
For y″ − 5y′ + 6y = 0 the auxiliary equation is m² − 5m + 6 = 0. What is its larger root? · 对 y″ − 5y′ + 6y = 0,辅助方程是 m² − 5m + 6 = 0。它较大的根是多少?
m² − 5m + 6 = (m − 2)(m − 3) = 0, so m = 2 or 3; the larger is 3. · m² − 5m + 6 = (m − 2)(m − 3) = 0,所以 m = 2 或 3;较大的是 3。
Match each type of auxiliary root to the complementary function it gives. · 把每种辅助根的类型匹配到它给出的补函数。
Distinct reals give two exponentials; a repeated root adds a factor of x; complex roots give an oscillating, exponentially-scaled form. · 相异的实数给出两个指数;一个重根添加一个 x 因子;复根给出一个振荡的、指数缩放的形式。
Boundary conditions
- The constants $A, B$ are fixed by boundary or initial conditions 初始条件 (e.g. $y(0) = 1$, $y'(0) = 0$).
- Without them you get the whole family; with them, the one curve that fits.
- Solve differential equations with a substitution or by separable variables, then apply initial conditions for the particular solution.
边界条件
- 常数 $A, B$ 由边界(boundary)或初始条件(initial conditions,例如 $y(0) = 1$,$y'(0) = 0$)确定。
- 没有它们你得到整个族;有了它们,得到匹配的那一条曲线。
- 用代换(substitution)或可分离变量(separable variables)解微分方程(differential equations),再用初始条件求特解(particular solution)。
The arbitrary constants A and B are fixed by the ______ conditions. · 任意常数 A 和 B 由______条件确定。
Boundary or initial conditions (like y(0) = 1) determine A and B. · 边界或初始条件(像 y(0) = 1)确定 A 和 B。
You've got it
- first-order linear: multiply by $\mu = e^{\int P\,dx}$, then integrate
- second-order: general = complementary function + particular integral 特积分
- the CF comes from the auxiliary equation; boundary conditions fix the constants
你掌握了
- 一阶线性:乘以 $\mu = e^{\int P\,dx}$,然后积分
- 二阶:通解 = 补函数 + 特解
- CF 来自辅助方程;边界条件确定常数