Differential Equations Key Points | 微分方程考点精讲

📚 Differential Equations Key Points | 微分方程考点精讲

A solid grasp of differential equations is essential for top marks in both IB Higher Level Mathematics (Analysis & Approaches or Applications & Interpretation) and OCR A Level Mathematics. This article distils the core techniques, common pitfalls, and exam strategies you need, from first‑order separable equations to second‑order non‑homogeneous ODEs and numerical methods.

扎实掌握微分方程是在 IB 高等级数学（分析与方法、应用与解释）和 OCR A Level 数学中取得高分的关键。本文浓缩了一阶可分离方程、二阶非齐次常微分方程以及数值方法等核心技巧、常见错误与应试策略。

1. Basic Concepts & Terminology | 基本概念与术语

A differential equation (DE) is an equation involving an unknown function and its derivatives. The order is the highest derivative present; the degree is the power of that highest derivative after the equation has been cleared of fractions and radicals. The general solution contains one arbitrary constant for a first‑order DE, two for a second‑order DE, and so on. A particular solution is obtained when initial or boundary conditions are applied to find those constants.

微分方程是包含未知函数及其导数的方程。阶数是方程中出现的最高阶导数；次数是消去分式与根式后该最高阶导数的幂。一阶微分方程的通解含有一个任意常数，二阶则含有两个，以此类推。代入初始条件或边界条件确定常数后便得到特解。

2. Separable First‑Order Equations | 可分离变量的一阶方程

If a first‑order ODE can be rearranged into the form g(y) dy = f(x) dx, it is called separable. Integrate both sides:

若一阶常微分方程可重排为 g(y) dy = f(x) dx 的形式，则称其为可分离变量方程。两边同时积分：

∫ g(y) dy = ∫ f(x) dx + C

Example: dy/dx = xy ⇒ (1/y) dy = x dx ⇒ ln|y| = ½ x² + C ⇒ y = A e^½x². Always consider whether y=0 is a lost solution when dividing by g(y).

例：dy/dx = xy ⇒ (1/y) dy = x dx ⇒ ln|y| = ½ x² + C ⇒ y = A e^½x²。当两边同除以 g(y) 时，务必检查 y=0 是否为被除掉的解。

3. First‑Order Linear Equations & Integrating Factor | 一阶线性方程与积分因子

The standard form is dy/dx + P(x) y = Q(x). The integrating factor (IF) is

标准形式为 dy/dx + P(x) y = Q(x)。积分因子为

μ(x) = e^{∫ P(x) dx}

Multiply the entire DE by μ(x). Then the left‑hand side becomes the derivative of μ(x)y exactly:

将整个方程乘以 μ(x)，左边恰好变为 μ(x)y 的导数：

d/dx [μ(x) y] = μ(x) Q(x)

Integrate both sides and solve for y. Example: dy/dx + 2y = eˣ ⇒ μ = e^2x ⇒ d/dx(e^2xy) = e^3x ⇒ e^2xy = ⅓ e^3x + C ⇒ y = ⅓ eˣ + C e^−2x.

两边积分后解出 y。例：dy/dx + 2y = eˣ ⇒ μ = e^2x ⇒ d/dx(e^2xy) = e^3x ⇒ e^2xy = ⅓ e^3x + C ⇒ y = ⅓ eˣ + C e^−2x。

4. Homogeneous First‑Order Equations (IB HL) | 齐次一阶方程（IB HL）

An ODE of the form dy/dx = F(y/x) can be tackled by the substitution v = y/x, i.e. y = vx. Differentiating gives dy/dx = v + x dv/dx. The resulting equation in v and x is always separable.

形如 dy/dx = F(y/x) 的方程可通过代换 v = y/x（即 y = vx）处理。求导得 dy/dx = v + x dv/dx，所得关于 v 与 x 的方程总是可分离变量的。

Example: dy/dx = (x+y)/x = 1 + y/x. Set v = y/x ⇒ v + x dv/dx = 1 + v ⇒ x dv/dx = 1 ⇒ dv = dx/x ⇒ v = ln|x| + C ⇒ y = x (ln|x| + C).

例：dy/dx = (x+y)/x = 1 + y/x。令 v = y/x ⇒ v + x dv/dx = 1 + v ⇒ x dv/dx = 1 ⇒ dv = dx/x ⇒ v = ln|x| + C ⇒ y = x (ln|x| + C)。

5. Second‑Order Linear Homogeneous ODEs with Constant Coefficients | 二阶常系数线性齐次方程

These take the form a d²y/dx² + b dy/dx + c y = 0. Assume a solution y = e^λx to obtain the auxiliary equation:

这类方程形如 a d²y/dx² + b dy/dx + c y = 0。设解为 y = e^λx，得到辅助方程：

a λ² + b λ + c = 0

Two distinct real roots λ₁, λ₂: y = A e^λ₁x + B e^λ₂x.
两个不等实根 λ₁, λ₂：y = A e^λ₁x + B e^λ₂x。
Repeated real root λ: y = (A + Bx) e^λx.
重实根 λ：y = (A + Bx) e^λx。
Complex conjugate roots α ± iβ: y = e^αx (A cos βx + B sin βx).
共轭复根 α ± iβ：y = e^αx (A cos βx + B sin βx)。

6. Non‑Homogeneous Second‑Order ODEs: Undetermined Coefficients | 非齐次二阶方程：待定系数法

For a d²y/dx² + b dy/dx + c y = f(x), the general solution is the sum of the complementary function (CF, solution of the homogeneous equation) and a particular integral (PI). The PI is found by making an educated guess based on f(x):

对于 a d²y/dx² + b dy/dx + c y = f(x)，通解为余函数（CF，即齐次方程的解）与特解（PI）之和。特解根据 f(x) 的形式合理猜测：

f(x)	Trial PI
polynomial of degree n	general polynomial of degree n
k e^px	C e^px
m cos ωx + n sin ωx	P cos ωx + Q sin ωx

If the trial PI already appears in the CF, multiply by x (or x² if necessary). Substitute back to determine unknown coefficients.

若猜测的特解形式已在 CF 中出现，则乘以 x（必要时乘以 x²）。代回原方程确定待定系数。

Example: y” − 3y’ + 2y = eˣ. CF: λ²−3λ+2=0 ⇒ λ=1,2 ⇒ y_c = A eˣ + B e^2x. Since eˣ is in the CF, try y_p = C x eˣ. After substitution, C = −1 ⇒ y = A eˣ + B e^2x − x eˣ.

例：y” − 3y’ + 2y = eˣ。CF：λ²−3λ+2=0 ⇒ λ=1,2 ⇒ y_c = A eˣ + B e^2x。因 eˣ 已在 CF 中，尝试 y_p = C x eˣ。代入求得 C = −1 ⇒ y = A eˣ + B e^2x − x eˣ。

7. Initial & Boundary Conditions | 初始条件与边界条件

To pin down a particular solution, you need as many conditions as the number of arbitrary constants. Substitute the given x₀, y₀ (and y'(x₀) for second‑order) into the general solution and its derivative, then solve the resulting simultaneous equations.

要确定特解，需要与任意常数个数相等的条件。将给定的 x₀, y₀（二阶时还包括 y'(x₀)）代入通解及其导数，然后解联立方程组即可。

Example: dy/dx = 2y, y(0)=5 ⇒ y = C e^2x ⇒ 5 = C e⁰ ⇒ C=5 ⇒ y = 5 e^2x.

例：dy/dx = 2y，y(0)=5 ⇒ y = C e^2x ⇒ 5 = C e⁰ ⇒ C=5 ⇒ y = 5 e^2x。

8. Application: Exponential Growth & Decay | 应用：指数增长与衰减

The ODE dy/dt = k y models natural growth (k>0) or decay (k<0). Its solution is y = y₀ e^kt. Classic contexts include unrestricted population growth, radioactive decay (half‑life), and continuously compounded interest.

微分方程 dy/dt = k y 描述了自然增长（k>0）或衰减（k<0），其解为 y = y₀ e^kt。经典场景包括无限制种群增长、放射性衰变（半衰期）以及连续复利。

Newton’s law of cooling follows a related pattern: dT/dt = −k (T − Tₐ), where Tₐ is the ambient temperature. Use separation of variables to find T(t).

牛顿冷却定律遵循类似模式：dT/dt = −k (T − Tₐ)，其中 Tₐ 为环境温度，用分离变量法可解出 T(t)。

9. Application: Kinematics & Mechanics | 应用：运动学与力学

In mechanics, Newton’s second law yields ODEs for displacement x(t) or velocity v(t). With constant acceleration, d²x/dt² = a, which is simply integrated. When resistive forces are proportional to velocity, you often obtain a first‑order linear ODE: m dv/dt = mg − kv, solvable via integrating factor.

在力学中，牛顿第二定律给出关于位移 x(t) 或速度 v(t) 的微分方程。匀加速时 d²x/dt² = a，直接积分即可。当阻力与速度成正比时，常得到一阶线性方程：m dv/dt = mg − kv，可用积分因子求解。

Example: A falling object experiences air resistance −kv. The terminal velocity is mg/k. Solve m dv/dt = mg − kv with v(0)=0 to obtain v = (mg/k)(1 − e^−kt/m).

例：落体受空气阻力 −kv 作用。终端速度为 mg/k。解 m dv/dt = mg − kv，v(0)=0，得 v = (mg/k)(1 − e^−kt/m)。

10. Numerical Methods: Euler’s Method | 数值方法：欧拉法

For dy/dx = f(x,y) with y(x₀)=y₀, Euler’s method approximates the solution by stepping forward with a fixed step size h:

对于 dy/dx = f(x,y)，y(x₀)=y₀，欧拉法通过固定步长 h 逐步递推近似解：

x_n+1 = x_n + h, y_n+1 = y_n + h f(x_n, y_n)

Smaller h gives better accuracy but requires more steps. This method is tested in IB HL (calculus option) and may appear in OCR’s numerical methods.

步长 h 越小精度越高，但计算步数增多。该方法在 IB HL（微积分选项）中经常考查，也可能出现在 OCR 的数值方法部分。

11. Common Mistakes & Exam Tips | 常见错误与考点提示

Forgetting the constant of integration – always write “+ C” immediately after integrating, and use initial conditions only after finding the general solution.
遗漏积分常数 – 积分后应立即写上“+ C”，务必先求通解再代入初始条件。
Integrating factor slip‑ups – rearrange into standard form dy/dx + P(x)y = Q(x) first; a common mistake is omitting the factor when multiplying Q(x).
积分因子操作失误 – 必须先化为标准形式 dy/dx + P(x)y = Q(x)；常见错误是乘 Q(x) 时漏乘积分因子。
Lost solutions in separation – when dividing by g(y), check whether g(y)=0 yields a valid constant solution.
分离变量时丢失解 – 当除以 g(y) 时，必须检查 g(y)=0 是否给出一个有效的常数解。
Wrong trial PI – ensure the guess does not duplicate a CF term; if it does, multiply by x (or x²).
特解猜测不当 – 确保猜测形式不与 CF 项重复；若重复则乘以 x（或 x²）。
Auxiliary equation root cases – be careful with repeated and complex roots; misidentifying leads to an incorrect CF.
辅助方程根的判别 – 小心处理重根与复根情形；误判会导致 CF 出错。

12. Summary & Revision Strategy | 总结与备考建议

Master the four core analytical methods: (1) separation of variables, (2) integrating factor, (3) homogeneous substitution, and (4) the auxiliary‑equation / undetermined‑coefficient pairing for second‑order ODEs. Work plenty of past‑paper problems from both IB and OCR to recognise the standard phrasing of modelling questions. During revision, summarise the ‘guess table’ for particular integrals and practise sketching slope fields where required. Finally, double‑check algebraic manipulation under exam pressure – a single sign error can derail the whole solution.

熟练掌握四种核心解析方法：(1) 分离变量法，(2) 积分因子法，(3) 齐次代换，以及 (4) 二阶方程的辅助方程＋待定系数法。大量练习 IB 与 OCR 的往年试题，熟悉建模题的标准表述。复习时总结特解的“猜测表格”，并在需要时练习绘制斜率场。最后，在考试压力下务必复查代数运算——一个符号错误就可能导致全盘皆输。

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Differential Equations Key Points | 微分方程 考点精讲