Integrating Factors: Solving First-Order Linear Differential Equations | 积分因子法：求解一阶线性微分方程

📚 Integrating Factors: Solving First-Order Linear Differential Equations | 积分因子法：求解一阶线性微分方程

any equation of the form dy/dx + P(x)y = Q(x), regardless of whether it is separable or exact. Mastering this technique is essential for success in IB Mathematics: Analysis and Approaches Higher Level as well as for further studies in engineering and physical sciences.

一阶线性常微分方程是最常见且实用性极强的微分方程类型，广泛应用于人口模型、电路分析和热传导问题。积分因子法提供了一种系统化方法，可求解所有形如 dy/dx + P(x)y = Q(x) 的方程，无论它是否是可分离或恰当方程。掌握这一技巧对于在 IB 数学分析与方法高水平考试中取得好成绩，以及未来在工程和自然科学领域深造都至关重要。

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1. What is a First-Order Linear ODE? | 什么是一阶线性常微分方程？

form dy/dx + P(x)y = Q(x), where P and Q are functions of x only (or constants). The term 'linear' irst power, with no products like y·(dy/dx) and no nonlinear functions such as sin(y) or eʸ.

一阶线性常微分方程是可以写成标准形式 dy/dx + P(x)y = Q(x) 的方程，其中 P 和 Q 只是 x 的函数（或常数）。“线性”指因变量 y 及其导数只以一次幂出现，没有 y·(dy/dx) 这样的乘积项，也没有 sin(y) 或 eʸ 等非线性函数。

dy/dx + P(x)y = Q(x)

If Q(x) = 0, the equation is called homogeneous; otherwise it is non‑homogeneous. The integrating factor method works for both the homogeneous and the non‑homogeneous cases in exactly the same way.

如果 Q(x) = 0，方程称为齐次方程；否则为非齐次方程。积分因子法以完全相同的方式处理齐次和非齐次两种情形。

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2. Standard Form | 标准形式

iding through by any coefficient in front of dy/dx and moving all terms involving y to one side.

在应用积分因子之前，你必须总是先把给定方程改写为标准形式 dy/dx + P(x)y = Q(x)。这通常意味着要除以 dy/dx 前的任何系数，并把所有含 y 的项移到一边。

2 to obtain dy/dx + 2y = 3x. Only then can you correctly identify P(x) = 2 and Q(x) = 3x.

例如，方程 2·dy/dx + 4y = 6x 必须先除以 2，得到 dy/dx + 2y = 3x。只有这样你才能正确识别出 P(x) = 2 和 Q(x) = 3x。

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3. The Integrating Factor Concept | 积分因子的概念

The idea behind the method is to multiply both sides of the standard form by a cleverly chosen function μ(x) – the integrating factor – so that the left‑hand side becomes the exact derivative of a product: d/dx(μ y). This transforms the original ODE into one that can be integrated directly.

该方法的核心思想是：用精心选取的函数 μ(x)——积分因子——去乘标准形式方程的两边，使得左边变成某个乘积的准确导数：d/dx(μ y)。这使原方程转化为可以直接积分的方程。

y. Comparing terms, this requires dμ/dx = μ P, which is a simple separable equation for μ.

我们希望有 μ·(dy/dx + P y) = d/dx(μ y)。根据乘积法则，d/dx(μ y) = μ·dy/dx + (dμ/dx)·y。比较各项可知，这要求 dμ/dx = μ P，这是一个关于 μ 的简单可分离方程。

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4. Deriving the Integrating Factor Formula | 积分因子公式的推导

rating variables in dμ/dx = μ P gives (1/μ) dμ = P(x) dx. Integrating both sides yields ln|μ| ) dx. Exponentiating, and choosing the positive branch for convenience, we obtain the well‑known formula for the integrating factor.

在 dμ/dx = μ P 中分离变量可得 (1/μ) dμ = P(x) dx。两边积分得到 ln|μ| = ∫ P(x) dx。取指数并为了方便选择正分支，就得到了著名的积分因子公式。

μ(x) = e^(∫ P(x) dx)

No constant of integration is needed when computing μ, because any constant multiple would cancel out when we multiply the whole equation. The simplest choice – taking the constant as zero – always works.

计算 μ 时不需要加积分常数，因为任何常数倍数在乘入整个方程时都会约掉。最简单的选择——取常数为零——总是有效的。

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5. Step-by-Step Method | 逐步求解法

The integrating factor method can be summarised in a clear sequence of steps. Following these steps carefully will solve any first‑order linear ODE provided the necessary integrals can be evaluated.

积分因子法可归纳为一套清晰的步骤。只要所需的积分可以求出，仔细遵循这些步骤就能解出任何一个一阶线性常微分方程。

Step 1	Write the equation in standard form dy/dx + P(x)y = Q(x).
Step 2	Identify P(x) and compute ∫ P(x) dx.
Step 3	Determine the integrating factor μ(x) = e^(∫ P(x) dx).
Step 4	Multiply every term of the standard form by μ(x).
Step 5	Recognise the left‑hand side as d/dx(μ y).
Step 6	Integrate both sides with respect to x: μ y = ∫ μ Q(x) dx + C.
Step 7	Solve for y, applying any initial conditions if given.

始终牢记：在积分因子法中，所有项都要乘以 μ，包括右边，并且一定要在积分后加上任意常数 C。

Always remember: in the integrating factor method, every term must be multiplied by μ, including the right‑hand side, and an arbitrary constant C must be added after integration.

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6. Example 1: Basic Equation | 示例1：基本方程

Solve dy/dx + 2y = e⁻ˣ.

求解 dy/dx + 2y = e⁻ˣ。

The equation is already in standard form with P(x) = 2 and Q(x) = e⁻ˣ. Compute ∫ P dx = ∫ 2 dx = 2x, thus μ = e²ˣ.

该方程已是标准形式，P(x) = 2，Q(x) = e⁻ˣ。计算 ∫ P dx = ∫ 2 dx = 2x，因此 μ = e²ˣ。

Multiply through by e²ˣ: e²ˣ·dy/dx + 2e²ˣ y = e²ˣ·e⁻ˣ = eˣ. The left side is d/dx(y e²ˣ). So d/dx(y e²ˣ) = eˣ.

用 e²ˣ 乘全式：e²ˣ·dy/dx + 2e²ˣ y = e²ˣ·e⁻ˣ = eˣ。左边为 d/dx(y e²ˣ)。所以 d/dx(y e²ˣ) = eˣ。

Integrate: y e²ˣ = ∫ eˣ dx + C = eˣ + C. Hence y = e⁻²ˣ(eˣ + C) = e⁻ˣ + C e⁻²ˣ.

积分：y e²ˣ = ∫ eˣ dx + C = eˣ + C。因此 y = e⁻²ˣ(eˣ + C) = e⁻ˣ + C e⁻²ˣ。

The general solution is y = e⁻ˣ + C e⁻²ˣ, where C is an arbitrary constant. You can verify by differentiating.

通解为 y = e⁻ˣ + C e⁻²ˣ，其中 C 为任意常数。你可以通过求导来验证。

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7. Example 2: Initial Value Problem | 示例2：初值问题

Solve x·dy/dx – y = x², given y(1) = 3.

求解 x·dy/dx – y = x²，已知 y(1) = 3。

First rewrite in standard form: divide by x (x > 0 assumed) to get dy/dx – (1/x) y = x. Here P(x) = –1/x, Q(x) = x.

首先改写成标准形式：除以 x（假定 x > 0）得 dy/dx – (1/x) y = x。这里 P(x) = –1/x，Q(x) = x。

Compute ∫ P dx = ∫ (–1/x) dx = –ln|x| = –ln x (since x>0). Then μ = e^{–ln x} = 1/x.

计算 ∫ P dx = ∫ (–1/x) dx = –ln|x| = –ln x（因 x>0）。则 μ = e^{–ln x} = 1/x。

Multiply the standard form by 1/x: (1/x)·dy/dx – (1/x²) y = 1. Left side is d/dx(y·(1/x)) = d/dx(y/x).

用 1/x 乘标准形式：(1/x)·dy/dx – (1/x²) y = 1。左边是 d/dx(y·(1/x)) = d/dx(y/x)。

So d/dx(y/x) = 1. Integrate: y/x = ∫ 1 dx + C = x + C. Thus y = x² + C x.

因此 d/dx(y/x) = 1。积分：y/x = ∫ 1 dx + C = x + C。于是 y = x² + C x。

Apply y(1)=3: 3 = 1² + C·1 → C = 2. The particular solution is y = x² + 2x.

代入 y(1)=3：3 = 1² + C·1 → C = 2。特解为 y = x² + 2x。

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8. Handling Absolute Values in Integration | 积分中绝对值的处理

When computing ∫ P(x) dx, you may encounter integrals of the form ∫ (1/x) dx = ln|x|. In theory, the absolute value ensures the domain is correct. However, when forming μ = e^{ln|x|⁺constant}, the exponential function maps real numbers to positive values, so the integrating factor can be taken as a positive function on any interval that does not include zero.

在计算 ∫ P(x) dx 时，你可能遇到形如 ∫ (1/x) dx = ln|x| 的积分。理论上，绝对值确保定义域正确。但在构成 μ = e^{ln|x|⁺常数} 时，指数函数将实数映射为正值，因此在任何不包含零的区间上，积分因子都可取为正函数。

In practice, for IB problems, the domain is usually restricted to x > 0, so you can drop the absolute value and write ln x. If the domain includes negative values, you must consider the sign, but the final solution often remains valid by absorbing a possible sign change into the arbitrary constant.

在实际的 IB 考题中，定义域通常限制在 x > 0，因此可以去掉绝对值写成 ln x。如果定义域包含负值，你必须考虑符号，但最终解通常可以通过将可能的符号变化吸收进任意常数而保持有效。

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9. Application: RL Circuit | 应用：RL电路

·di/dt + R i = E(t). Rewriting in standard form: di/dt + (R/L) i = E(t)/L. This is a first‑order linear ODE with P(t) = R/L (constant).

一个含有电感 L、电阻 R 和电压源 E(t) 的 RL 电路可用 L·di/dt + R i = E(t) 描述。改写成标准形式：di/dt + (R/L) i = E(t)/L。这是一个一阶线性常微分方程，其中 P(t) = R/L（常数）。

The integrating factor is μ = e^{∫ R/L dt} = e^Rt/L. Multiplying through by μ gives d/dt(i e^Rt/L) = (E(t)/L) e^Rt/L. Integrating and solving for i yields the complete current response.

积分因子为 μ = e^{∫ R/L dt} = e^Rt/L。用 μ 乘全式得到 d/dt(i e^Rt/L) = (E(t)/L) e^Rt/L。积分并解出 i 就可得到完整的电流响应。

For a constant voltage E(t) = E₀, the solution becomes i(t) = E₀/R + C e^–Rt/L. The term C e^–Rt/L represents the transient current that decays exponentially, while E₀/R is the steady‑state current.

对于恒定电压 E(t) = E₀，解为 i(t) = E₀/R + C e^–Rt/L。其中 C e^–Rt/L 项表示随时间指数衰减的暂态电流，而 E₀/R 是稳态电流。

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10. Common Mistakes | 常见错误

One frequent error is forgetting to put the equation into standard form before identifying P(x). If the coefficient of dy/dx is not 1, the wrong P will lead to an incorrect integrating factor.

一个常见错误是在识别 P(x) 之前忘记把方程化为标准形式。如果 dy/dx 的系数不是 1，错误的 P 将导致错误的积分因子。

Another mistake is missing the arbitrary constant C after integration. Without it, only one particular solution is captured, and initial conditions cannot be satisfied. Also, students sometimes forget to multiply the right‑hand side Q(x) by the integrating factor, which destroys the equality.

另一个错误是积分后漏掉了任意常数 C。缺少它就只能得到一个特解，且无法满足初值条件。此外，学生有时忘记把右边 Q(x) 也乘以积分因子，这破坏了等式的正确性。

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11. Practice Tips | 练习建议

To build confidence, practise recognising the standard form quickly: scan for the derivative term and isolate it before anything else. Always write P(x) and Q(x) explicitly on the side of your page.

为增强信心，要练习快速识别标准形式：先扫视导数项并将其隔离。始终在页面边角位置明确写出 P(x) 和 Q(x)。

P(x) is not a simple constant, such as P(x) = tan x or 2/x, to master integration skills.

找到解之后，求导并将其代回原方程作为检验。这一验证步骤快捷有效，能发现代数错误。同时，要尝试 P(x) 不是简单常数的问题，例如 P(x) = tan x 或 2/x，以锻炼积分能力。

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12. Summary | 总结

The integrating factor technique transforms a first‑ and the sequence of rewriting, multiplying, recognising the derivative of the product, and integrating.

积分因子技术将一阶线性常微分方程转化为一个恰当导数，把问题归结为计算两个积分。该方法围绕公式 μ = e^(∫ P(x) dx) 展开，步骤依次为改写、相乘、识别乘积导数以及积分。

With disciplined practice, you can solve any IB examination question on this topic reliably and efficiently, laying a strong foundation for more advanced differential equations in future studies.

通过有纪律的练习，你能够可靠高效地解决该专题的任何 IB 考题，并为日后学习更高阶的微分方程打下坚实基础。

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