📚 Differential Equations and Their Solutions | 微分方程及其解法
Differential equations form a cornerstone of calculus and mathematical modelling. They connect rates of change with functions themselves, allowing us to describe everything from population growth and radioactive decay to mechanical vibrations and electrical circuits. In the IB Mathematics programme, you are expected to solve first-order separable and linear equations, recognise homogeneous types, and handle second-order linear equations with constant coefficients. You must also verify solutions by substitution and interpret initial conditions to find particular solutions.
微分方程是微积分和数学建模的基石。它们将变化率与函数本身联系起来，使我们能够描述从人口增长、放射性衰变到机械振动和电路的各种现象。在IB数学课程中，你需要求解一阶可分离变量方程和线性方程，识别齐次类型，并处理二阶常系数线性方程。你还必须通过代入来验证解，并利用初始条件求特解。
1. What Is a Differential Equation? | 什么是微分方程？
nds on a single independent variable, the equation is called an ordinary differential equation (ODE). If it involves partial derivatives, it is a partial differential equation, though only ODEs appear in the IB syllabus.
微分方程是包含未知函数及其一个或多个导数的方程。例如，dy/dx = 3x² 是一个简单的微分方程，其解为 y = x³ + C。如果未知函数只依赖于一个自变量，该方程称为常微分方程（ODE）。若涉及偏导数，则是偏微分方程，不过IB大纲只涉及常微分方程。
- dy/dx = 2x + y
- d²y/dx² – 5dy/dx + 6y = 0
- y'' + 2y' + y = eˣ
- dy/dx = 2x + y
- d²y/dx² – 5dy/dx + 6y = 0
- y'' + 2y' + y = eˣ
2. Order and Degree | 阶与次
t to derivatives. For instance, (d²y/dx²)² + (dy/dx)³ = 1 is second-order and of degree 2.
微分方程的阶是指方程中出现的最高阶导数的阶数。dy/dx = x² 是一阶；d²y/dx² + 3dy/dx = 0 是二阶。次数则是在将方程对导数去除根式和分式幂之后，最高阶导数的指数。例如 (d²y/dx²)² + (dy/dx)³ = 1 是二阶二次。
ritten as dy/dx + P(x)y = Q(x). Any equation that cannot be put in this form is nonlinear.
IB考试题常要求你识别方程的阶，并将其分为线性或非线性。一阶线性ODE可以写成 dy/dx + P(x)y = Q(x) 的形式。不能写成这种形式的方程就是非线性的。
3. General and Particular Solutions | 通解与特解
he constant; the result is a particular solution. The extra condition is called an initial condition or boundary condition.
n阶常微分方程的通解包含n个任意常数。例如 dy/dx = 2x 的通解是 y = x² + C。在应用中，额外的信息（如当 x = 0 时 y 的值）使我们能够确定常数，所得结果就是特解。这个额外条件称为初始条件或边界条件。
To verify that a function is a solution, differentiate it as required and substitute into the original equation. If the left-hand side reduces identically to the right-hand side, the function satisfies the ODE.
要验证一个函数是否是解，按需求对其求导并代入原方程。如果左边恒等于右边，那么该函数就满足该常微分方程。
4. Separable Differential Equations | 可分离变量微分方程
yields the general solution. This technique is fundamental and appears frequently in IB examinations.
如果一阶常微分方程可以写成 dy/dx = f(x)g(y) 的形式，称其为可分离变量的。变量可以分离：1/g(y) dy = f(x) dx。两边积分便得到通解。这一基础技巧在IB考试中频繁出现。
Step 1: rearrange to 1/g(y) dy = f(x) dx
第1步：整理成 1/g(y) dy = f(x) dx
Step 2: ∫1/g(y) dy = ∫f(x) dx
第2步：∫1/g(y) dy = ∫f(x) dx
Example: solve dy/dx = 2xy. Separate to 1/y dy = 2x dx, integrate to ln|y| = x² + C, so y = Aeˣ² where A = eᶜ.
例题：求解 dy/dx = 2xy。分离变量得 1/y dy = 2x dx，积分得 ln|y| = x² + C，因此 y = Aeˣ²，其中 A = eᶜ。
question provides an initial condition, such as y(0) = 3, which allows you to calculate A.
务必记得立即加上积分常数。有时题目会给出初始条件，例如 y(0) = 3，据此可算出常数 A。
5. First-Order Linear Differential Equations | 一阶线性微分方程
(here meaning no forcing term, but not to be confused with the homogeneous substitution method).
一阶线性常微分方程的标准形式为 dy/dx + P(x)y = Q(x)。求解它需要一个积分因子。该方程对 y 及其导数是线性的；Q(x) 称为驱动项。如果 Q(x) = 0，方程为齐次的（此处指无驱动项，注意勿与齐次代换法混淆）。
IB problems often present equations that are not instantly recognisable in this form. You must manipulate the equation into the standard linear structure before identifying P(x) and Q(x).
IB题目中给出的方程往往无法直接识别出这种形式。你需要先将方程整理为标准线性结构，再确定 P(x) 和 Q(x)。
6. Integrating Factor Method | 积分因子法
Multiply the entire equation by μ(x). The left side becomes the derivative of μ(x)y. Then integrate both sides with respect to x and solve for y.
对于 dy/dx + P(x)y = Q(x)，计算积分因子 μ(x) = e^∫P(x) dx。将整个方程乘以 μ(x)，左边即变为 μ(x)y 的导数。然后两边对 x 积分，并解出 y。
μ(x) = e^∫P(x) dx
μ(x) = e^∫P(x) dx
d/dx [μ(x)y] = μ(x)Q(x)
d/dx [μ(x)y] = μ(x)Q(x)
Example: solve dy/dx + y/x = x², x > 0. Here P(x)=1/x, so ∫P(x)dx = ln x, μ(x)= eln x by x: x dy/dx + y = x³. Left is d/dx(xy), thus xy = ∫x³ dx = x⁴/4 + C, and y = x³/4 + C/x.
例题：求解 dy/dx + y/x = x²，x > 0。这里 P(x)=1/x，因此 ∫P(x)dx = ln x，μ(x) = eln x = x。乘以 x 得：x dy/dx + y = x³。左边即 d/dx(xy)，于是 xy = ∫x³ dx = x⁴/4 + C，y = x³/4 + C/x。
Remember that the IB often sets problems where you must find the constant using an initial condition. Substitute x and y immediately after obtaining the general solution to find the particular solution.
记住，IB时常要求利用初始条件求常数。在得到通解后立即代入 x 和 y，以求出特解。
7. Homogeneous Differential Equations (by Substitution) | 齐次微分方程（通过代换）
+ x dv/dx. This transforms the equation into a separable one in v and x. Solve it, then replace v by y/x to get the solution.
若一阶常微分方程可写成 dy/dx = F(y/x) 的形式，则称其为 n 次齐次方程。使用代换 v = y/x，则 y = vx，且 dy/dx = v + x dv/dx。由此将原方程转化为关于 v 与 x 的可分离变量方程。解出之后，再用 y/x 替换 v 即得解。
v = y/x ⇒ dy/dx = v + x dv/dx
v = y/x ⇒ dy/dx = v + x dv/dx
rite as dy/dx = 1 + (y/x). Substitute v = y/x, dy/dx = v + x dv/dx, so v + x dv/dx = 1 + v ⇒ x dv/dx = 1. Separate: dv = dx/x, integrate: v = ln|x| + C, hence y/x = ln|x| + C, y = x ln|x| + Cx.
例题：dy/dx = (x + y)/x。改写为 dy/dx = 1 + (y/x)。令 v = y/x，dy/dx = v + x dv/dx，于是 v + x dv/dx = 1 + v ⇒ x dv/dx = 1。分离变量：dv = dx/x，积分得 v = ln|x| + C，故 y/x = ln|x| + C，y = x ln|x| + Cx。
In IB mark schemes, showing the substitution and the subsequent separation of variables is essential for full credit. Always discard the v substitution appropriately in the final answer.
在IB评分标准中，展示代换步骤以及随后的变量分离过程对于拿到满分至关重要。最后答案一定要将 v 重新换回包含 y 和 x 的形式。
8. Second-Order Linear Homogeneous Equations with Constant Coefficients | 二阶常系数齐次线性方程
stants and a ≠ 0. These equations model oscillatory systems, damping, and many physical processes. The solution method relies on the characteristic equation.
二阶常系数线性常微分方程的形式为 a d²y/dx² + b dy/dx + c y = 0，其中 a、b、c 为常数且 a ≠ 0。这类方程可模拟振动系统、阻尼以及许多物理过程。求解方法依赖于特征方程。
To solve, assume a trial solution y = eʳˣ uadratic ar² + br + c = 0. The roots r₁ and r₂ determine the form of the general solution.
求解时，假设试解 y = eʳˣ。代入后得到二次方程 ar² + br + c = 0。根 r₁ 和 r₂ 决定了通解的形式。
9. Characteristic Equation and Its Roots | 特征方程及其根
Given ar² + br + c = 0, three cases arise.
给定 ar² + br + c = 0，会出现三种情况。
| Roots | General Solution |
| Real distinct r₁ ≠ r₂ | y = Aer₁x + Ber₂x |
| Real repeated r₁ = r₂ = r | y = (A + Bx)erx |
| Complex α ± iβ | y = eαx[A cos(βx) + B sin(βx)] |
| 根的情况 | 通解形式 |
| 相异实根 r₁ ≠ r₂ | y = Aer₁x + Ber₂x |
| 相等实根 r₁ = r₂ = r | y = (A + Bx)erx |
| 复根 α ± iβ | y = eαx[A cos(βx) + B sin(βx)] |
Example: y'' – 3y' + 2y = 0 → r² – 3r + 2 = 0 → r = 1, 2 → y = Aeˣ + Be²ˣ. If initial conditions y(0)=2, y'(0)=3 are given, substitute to find A and B.
例题：y'' – 3y' + 2y = 0 → r² – 3r + 2 = 0 → r = 1, 2 → y = Aeˣ + Be²ˣ。若给出初始条件 y(0)=2, y'(0)=3，代入即可求出 A 和 B。
IB questions often require you to construct the particular solution from two initial conditions. Write the system of equations for A and B clearly.
IB题目时常要求你由两个初始条件构造特解。请清晰地列出关于 A 和 B 的方程组。
10. Verification of Solutions | 解的验证
You may be asked to show that a given function satisfies a differential equation. Substitute y into the left-hand side, compute derivatives, and simplify algebraically until you match the right-hand side. This is commonly tested to confirm understanding of the chain rule, product rule, and implicit differentiation.
你可能需要证明某个给定函数满足微分方程。将 y 代入左边，求导，并通过代数化简直至与右边一致。这一过程常用来检测你对链式法则、乘法法则以及隐函数求导的理解。
For example, verify that y = x sin x is a solution of x² y'' – 2x y' + (x² + 2)y = 0. Compute y' = sin x + x cos x, y'' = 2 cos x – x sin x, then substitute and simplify to 0.
例如，验证 y = x sin x 是 x² y'' – 2x y' + (x² + 2)y = 0 的解。计算 y' = sin x + x cos x，y'' = 2 cos x – x sin x，然后代入并化简至0。
Verification is an excellent way to check your own answers during revision. If you derive a particular solution, plugging it back into the original ODE confirms its correctness.
验证是在复习时检查自己答案的好方法。如果你求出了一个特解，将其代回原常微分方程可以确认其正确性。
11. Applications: Exponential Growth and Decay | 应用：指数增长与衰减
, it describes growth; if k < 0, decay. Common contexts include population models, radioactive decay (half-life), and continuously compounded interest.
许多现实世界的过程都可以用可分离方程 dy/dt = ky 来建模，其解为 y = y₀ eᵏᵗ。若 k > 0，描述的是增长；若 k < 0，则是衰减。常见的例子包括种群模型、放射性衰变（半衰期）以及连续复利。
IB problems will often give a doubling time or half-life and ask you to find k, then answer follow-up questions such as the value of y at a specific time. The tangent proportional to the quantity is the defining characteristic of exponential change.
IB试题通常会给出倍增时间或半衰期，要求你求出 k，然后回答诸如在某特定时刻 y 的值等问题。切线与量成正比是指数变化的核心特征。
Newton’s law of cooling and mixing problems also lead to first-order linear differential equations and are part of the applied section of the syllabus.
牛顿冷却定律和混合问题同样会导出一阶线性微分方程，它们是大纲中应用部分的内容。
12. Summary and Exam Tips | 总结与考试技巧
- Identify the type of ODE before choosing a method (separable, linear, homogeneous substitution, or constant-coefficient second-order).
- Always write the standard form for linear equations and compute the integrating factor carefully.
- When separating variables, check that division by g(y) is valid; discuss singular solutions if required.
- For homogeneous substitution, explicitly state v = y/x and dy/dx = v + x dv/dx.
- For second-order ODEs, construct the characteristic equation, factor or use the quadratic formula, and write the general solution following the three cases.
- Use initial conditions to find constants, and verify your final answer by substitution.
- Maintain careful notation with derivatives; avoid rounding intermediate values in applied problems.
- 在选择解法之前，先识别常微分方程的类型（可分离、线性、齐次代换或常系数二阶）。
- 始终将线性方程写为标准形式，并仔细计算积分因子。
- 分离变量时，检查除以 g(y) 是否有效；若需要，讨论奇解。
- 对于齐次代换，明确写出 v = y/x 以及 dy/dx = v + x dv/dx。
- 对于二阶常微分方程，构造特征方程，使用因式分解或求根公式，并按照三种情况写出通解。
- 利用初始条件求常数，再通过代入来验证最终答案。
- 保持导数符号书写规范；在应用问题中，避免对中间值进行舍入。
Master these techniques, and differential equations will become one of the most predictable and rewarding topics on your IB Mathematics exam.
掌握这些方法，微分方程将成为IB数学考试中最具可预测性且回报最高的专题之一。
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