📚 Using Substitution y = vx to Solve Homogeneous Differential Equations | 使用代换 y = vx 求解齐次微分方程
In IB Mathematics Analysis and Approaches HL, solving first-order differential equations often requires identifying special forms that can be reduced to separable equations. One of the most powerful techniques is the substitution y = vx, which transforms a homogeneous differential equation into a separable one. This method works because replacing y with vx exploits the scaling properties of homogeneous functions, turning the original equation into a relationship between v and x that can be integrated directly. This article explores the theory behind the substitution, walks through detailed examples, and highlights common pitfalls and exam strategies to help you master this crucial topic.
在IB数学分析与方法HL课程中，求解一阶微分方程常常需要识别出那些可以化为可分离变量的特殊形式。最有力的技巧之一就是使用代换 y = vx，它能把齐次微分方程转化为可分离变量方程。这个方法之所以有效，是因为用 vx 替换 y 利用了齐次函数的缩放性质，将原方程转化为仅含 v 和 x 的关系式，从而可以直接积分。本文将深入探讨这一代换背后的理论，通过详细例题逐步演示，并指出常见错误与考试策略，帮助你彻底掌握这一关键内容。
1. What Is a Homogeneous Differential Equation? | 什么是齐次微分方程？
A first-order differential equation written as dy/dx = F(x, y) is called homogeneous if the function F(x, y) satisfies the condition F(tx, ty) = F(x, y) for any non-zero scalar t. In other words, F is a homogeneous function of degree zero. In practice, this means that every term in the expression for dy/dx has the same total degree when you count powers of x and y together. Look for forms like (x² + y²)/(2xy), (y/x) + tan(y/x), or (x³ + y³)/(xy²), where each component’s total degree cancels out to leave a function that depends only on the ratio y/x or x/y.
形如 dy/dx = F(x, y) 的一阶微分方程，若对任意非零标量 t 都有 F(tx, ty) = F(x, y)，则称其为齐次方程。换言之，F 是一个零次齐次函数。在实际操作中，这意味着 dy/dx 表达式中的每一项在统计 x 和 y 的总次数时都是相同的。常见的形式有 (x² + y²)/(2xy)、(y/x) + tan(y/x) 或 (x³ + y³)/(xy²)，这些分式中的总次数相互抵消，最终函数只依赖于比值 y/x 或 x/y。
Testing for homogeneity is the crucial first step. Replace x by tx and y by ty in F(x, y) and simplify. If all t’s cancel, the equation is homogeneous and the substitution y = vx is applicable.
检验齐次性是关键的第一步。将 F(x, y) 中的 x 替换为 tx，y 替换为 ty 并化简。如果所有的 t 都消去了，那么方程就是齐次的，可以使用代换 y = vx。
2. The Key Idea of Substitution y = vx | 代换 y = vx 的核心思想
When y is replaced by vx, the variable y is decomposed into two factors: x and a new dependent variable v (v is a function of x). This substitution guarantees that any zero-degree homogeneous function F(x, y) becomes a function of v alone, because the x-dependence factors out and cancels. The derivative dy/dx is transformed using the product rule: since y = vx, we have dy/dx = v + x dv/dx. The original equation dy/dx = F(x, y) then becomes v + x dv/dx = G(v), where G(v) is obtained from F after canceling x. This is a separable equation in v and x, which can be rearranged to (1/(G(v)-v)) dv = (1/x) dx, provided G(v) ≠ v.
当用 vx 替换 y 时，变量 y 被分解为两个因子：x 和一个新的因变量 v（v 是 x 的函数）。这种代换保证了任何零次齐次函数 F(x, y) 会变成仅含 v 的函数，因为 x 的依赖项会被因式分解并抵消。导数 dy/dx 则需利用乘积法则转换：由于 y = vx，得到 dy/dx = v + x dv/dx。原方程 dy/dx = F(x, y) 于是变为 v + x dv/dx = G(v)，其中 G(v) 是将 F 中的 x 约分后得到的关于 v 的表达式。这就变成了一个关于 v 和 x 的可分离变量方程，可以整理为 (1/(G(v)-v)) dv = (1/x) dx，前提是 G(v) ≠ v。
The beauty of this approach is its systematic nature: every homogeneous equation reduces to an integral of the form ∫ f(v) dv = ∫ (1/x) dx = ln|x| + C, which can then be solved by standard integration techniques.
这种方法的精妙之处在于其系统性：每个齐次方程最终都能化为形如 ∫ f(v) dv = ∫ (1/x) dx = ln|x| + C 的积分，然后用标准积分方法求解。
3. Step-by-Step Procedure | 逐步操作流程
To apply the substitution effectively, follow this structured approach:
为有效使用这个代换，请遵循以下结构化步骤：
- Verify homogeneity: Check that dy/dx can be written as a function of y/x or x/y with no leftover x or y terms.
验证齐次性：检查 dy/dx 能否写成 y/x 或 x/y 的函数且不剩余单独的 x 或 y 项。 - Set up the substitution: Let v = y/x, so y = vx and dy/dx = v + x dv/dx.
设立代换：令 v = y/x，于是 y = vx 且 dy/dx = v + x dv/dx。 - Substitute into the equation: Replace y and dy/dx with the expressions in terms of v and x. Simplify until the equation becomes v + x dv/dx = G(v).
代入方程：将 y 和 dy/dx 替换为关于 v 和 x 的表达式，化简直至得到 v + x dv/dx = G(v)。 - Separate the variables: Rearrange to x dv/dx = G(v) – v, then separate as (1/(G(v)-v)) dv = (1/x) dx.
分离变量：移项得 x dv/dx = G(v) – v，然后分离变量为 (1/(G(v)-v)) dv = (1/x) dx。 - Integrate both sides: Find ∫ (1/(G(v)-v)) dv = ∫ (1/x) dx. The right side gives ln|x| + C.
两边积分：计算 ∫ (1/(G(v)-v)) dv = ∫ (1/x) dx，右侧得 ln|x| + C。 - Back-substitute: Replace v by y/x to obtain the general solution in terms of x and y.
回代：将 v 替换回 y/x，得到用 x 和 y 表示的通解。 - Apply initial conditions (if given): Use given values to find the constant C and, if needed, simplify the solution.
应用初始条件（如给出）：利用所给数值求出常数 C，必要时化简解的形式。
4. Example 1: A Simple Homogeneous Equation | 例1：简单齐次方程
Consider the differential equation dy/dx = (x² + y²) / (2xy). First, test homogeneity: replace x with tx and y with ty to get (t²x² + t²y²) / (2·tx·ty) = t²(x²+y²) / (2t²xy) = (x²+y²)/(2xy), independent of t. So it is homogeneous.
考虑微分方程 dy/dx = (x² + y²) / (2xy)。首先检验齐次性：将 x 替换为 tx，y 替换为 ty，得 (t²x² + t²y²) / (2·tx·ty) = t²(x²+y²) / (2t²xy) = (x²+y²)/(2xy)，不含 t，因此是齐次方程。
Let y = vx, then dy/dx = v + x dv/dx. Substitute into the right side: (x² + v²x²)/(2x·vx) = x²(1+v²)/(2vx²) = (1+v²)/(2v). The equation becomes:
令 y = vx，则 dy/dx = v + x dv/dx。代入右侧：(x² + v²x²)/(2x·vx) = x²(1+v²)/(2vx²) = (1+v²)/(2v)。方程化为：
v + x dv/dx = (1 + v²)/(2v)
Subtract v: x dv/dx = (1+v²)/(2v) – v = (1+v² – 2v²)/(2v) = (1 – v²)/(2v). Separation of variables yields:
两边减去 v：x dv/dx = (1+v²)/(2v) – v = (1+v² – 2v²)/(2v) = (1 – v²)/(2v)。分离变量得：
(2v)/(1 – v²) dv = (1/x) dx
Integrate both sides: ∫ 2v/(1 – v²) dv = ∫ (1/x) dx. Let u = 1 – v², du = –2v dv, so the left integral becomes –∫ (1/u) du = –ln|u| = –ln|1 – v²|. Thus:
两边积分：∫ 2v/(1 – v²) dv = ∫ (1/x) dx。令 u = 1 – v²，du = –2v dv，左侧积分变为 –∫ (1/u) du = –ln|u| = –ln|1 – v²|。于是：
–ln|1 – v²| = ln|x| + C
Multiply by –1 and combine logs: ln|1/(1 – v²)| = ln|C₁x|, where C₁ = e⁻ᶜ. So 1/(1 – v²) = ±C₁ x. Let A be the arbitrary constant, giving 1/(1 – v²) = A x. Replace v = y/x: 1/(1 – y²/x²) = A x → x²/(x² – y²) = A x. Cancel one x (assuming x ≠ 0): x/(x² – y²) = A, so:
两边乘以 –1 并合并对数：ln|1/(1 – v²)| = ln|C₁x|，其中 C₁ = e⁻ᶜ。因此 1/(1 – v²) = ±C₁ x。令 A 为任意常数，得 1/(1 – v²) = A x。代回 v = y/x：1/(1 – y²/x²) = A x → x²/(x² – y²) = A x。约去一个 x（设 x ≠ 0）：x/(x² – y²) = A，于是：
y² = x² – (1/A) x
Renaming the constant, the general solution is y² = x² – C x, where C is an arbitrary constant.
重新命名常数，通解为 y² = x² – C x，其中 C 为任意常数。
5. Example 2: Equation with Trigonometric Terms | 例2：含三角函数的方程
Solve dy/dx = y/x + tan(y/x). Clearly, the right-hand side is a function of y/x alone, so it is homogeneous. Set v = y/x → y = vx, dy/dx = v + x dv/dx. The equation becomes:
求解 dy/dx = y/x + tan(y/x)。显然，右侧仅依赖于 y/x，因此是齐次方程。令 v = y/x → y = vx，dy/dx = v + x dv/dx。方程变为：
v + x dv/dx = v + tan v
Cancel v from both sides: x dv/dx = tan v. Separate variables: cot v dv = (1/x) dx. Integrate: ∫ cot v dv = ∫ (1/x) dx → ln|sin v| = ln|x| + C. Exponentiate:
两边消去 v：x dv/dx = tan v。分离变量：cot v dv = (1/x) dx。积分：∫ cot v dv = ∫ (1/x) dx → ln|sin v| = ln|x| + C。指数化得：
sin v = C x
(The constant absorbs the ± sign.) Substitute back v = y/x: sin(y/x) = C x. This implicit solution represents the general solution. If an initial condition were given, plug in the values to find C.
（常数吸收了 ± 号。）代回 v = y/x：sin(y/x) = C x。这个隐式解即为通解。若给出初始条件，代入数值即可求出 C。
6. Handling Initial Conditions | 处理初始条件
In IB problems, you will often be asked to find a particular solution. After obtaining the general solution, substitute the given x₀ and y₀ to determine the constant. For instance, using the previous example with condition y(1) = π/2, plug x=1, y=π/2 into sin(y/x) = C x: sin(π/2) = C·1 → 1 = C. Hence the particular solution is sin(y/x) = x, or explicitly y = x arcsin(x), with domain restrictions.
在IB题目中，经常会要求求出特解。得到通解后，代入给定的 x₀ 和 y₀ 来确定常数。例如，对上例施加条件 y(1) = π/2，代入 x=1, y=π/2 到 sin(y/x) = C x：sin(π/2) = C·1 → 1 = C。因此特解为 sin(y/x) = x，或显式写作 y = x arcsin(x)，并注意定义域限制。
Always check that your particular solution satisfies the original differential equation and that the constant does not lead to division by zero or invalid domains. If the integration involved ln|…|, be careful with absolute values when plugging in numbers.
始终检查你的特解是否满足原微分方程，并确保常数不会导致分母为零或定义域无效。如果积分过程中涉及 ln|…|，代入数值时要小心绝对值的影响。
7. Verifying Homogeneity Before Substitution | 代换前验证齐次性
A common mistake is applying y = vx to a non-homogeneous equation. Always rewrite the equation in the standard derivative form and examine whether replacing x with tx and y with ty leaves F unchanged. This can be done quickly by factoring out the highest powers and checking if all t’s cancel. If they do not, another method (such as integrating factor or substitution of a linear form) must be used. For example, dy/dx = (x + y + 1)/(x – y + 2) is not homogeneous because of the constant terms. Those need a shift of origin to be reduced to homogeneous form.
一个常见错误是错误地对非齐次方程使用 y = vx 代换。务必先将方程写为标准导数形式，并检查用 tx 替换 x、ty 替换 y 后 F 是否保持不变。可以通过提取最高次幂并观察所有 t 是否消去来快速判断。如果不能消去，则必须使用其他方法（如积分因子或线性形式代换）。例如，dy/dx = (x + y + 1)/(x – y + 2) 因为含有常数项而不是齐次方程，需要通过坐标平移才能化为齐次形式。
Also, note that some equations may appear homogeneous after simple algebraic manipulation. For instance, xy dy/dx = x² + y² can be rearranged to dy/dx = (x² + y²)/(xy), which is homogeneous. Always isolate dy/dx before testing.
还需注意，有些方程经过简单的代数操作后可能会呈现出齐次形式。例如，xy dy/dx = x² + y² 可整理成 dy/dx = (x² + y²)/(xy)，这就是齐次的。检验前务必先将 dy/dx 单独分离出来。
8. Common Pitfalls and How to Avoid Them | 常见陷阱及避免方法
- Forgetting the product rule: In substituting y = vx, many students write dy/dx = x dv/dx instead of v + x dv/dx. Always remember you are differentiating a product.
忘记乘积法则：在使用 y = vx 时，许多学生错误地写成 dy/dx = x dv/dx，遗漏了 v 这一项。务必记住你是在对乘积求导。 - Algebraic errors during separation: When moving terms, double-check the step x dv/dx = G(v) – v. After subtracting v, ensure you combine fractions correctly.
分离变量时的代数错误：移项时，仔细核对 x dv/dx = G(v) – v 这一步。减去 v 后，确保正确合并分式。 - Misplacing absolute values in logs: Integrals of 1/x and many rational functions produce ln|expression|. Failing to use absolute values can lose valid solutions, especially when later imposing initial conditions.
对数中的绝对值位置不当：对 1/x 以及许多有理函数的积分会产生 ln|表达式|。不使用绝对值可能会丢失有效解，尤其是在后面施加初始条件时。 - Not simplifying the final constant: The constant of integration often appears in a form like eᶜ or ±eᶜ. Rename it as a new arbitrary constant (say A or K) and state clearly that it can be any non-zero real number (or zero if allowed by the context).
未简化最终常数：积分常数经常以 eᶜ 或 ±eᶜ 的形式出现。将其重新命名为新的任意常数（如 A 或 K），并清楚地说明它可以取任意非零实数（或上下文允许的零）。 - Ignoring the singular solution: When you separate by dividing by G(v) – v, you assume that G(v) – v ≠ 0. The case G(v) – v = 0 may yield singular solutions (i.e., v = constant, leading to y = k x). Check if these satisfy the original equation.
忽略奇解：通过除以 G(v) – v 来分离变量时，你假定了 G(v) – v ≠ 0。G(v) – v = 0 的情形可能会产生奇解（即 v = 常数，从而 y = k x）。检查它们是否满足原方程。
9. Why Substitution Works – The Underlying Principle | 代换为何有效——基本原理
The substitution y = vx is rooted in the concept of homogeneous functions. A function f(x, y) is homogeneous of degree n if f(tx, ty) = tⁿ f(x, y). When F(x, y) in dy/dx = F(x, y) is of degree 0, scaling x and y by the same factor doesn’t change its value. This implies that F can be expressed purely in terms of the ratio y/x (or x/y). Let v = y/x; then y = vx, and any zero-degree homogeneous function F(x, y) simplifies to F(1, v) or F(1/v, 1). This transformation turns the original equation into a single-variable context, reducing the problem to a separable equation.
代换 y = vx 深植于齐次函数的概念。若 f(tx, ty) = tⁿ f(x, y)，则称 f(x, y) 是 n 次齐次函数。当 dy/dx = F(x, y) 中的 F 是 0 次时，同时缩放 x 和 y 不会改变其值。这意味着 F 可以纯粹用比值 y/x（或 x/y）表达。令 v = y/x，则 y = vx，任何零次齐次函数 F(x, y) 都会简化为 F(1, v) 或 F(1/v, 1)。这一变换将原方程化为单变量语境，将问题约简为可分离变量方程。
Geometrically, the substitution aligns with the symmetry of direction fields. The slope at (x, y) depends only on the angle from the origin, making the solution curves similar under radial scaling. Hence a change of variable that “factors out” the radial distance leaves an equation that is integrable in the angular variable v.
从几何上看，这一代换与方向场的对称性一致。点 (x, y) 处的斜率只依赖于其与原点的夹角，使得解曲线在径向缩放变换下相似。因此，将径向距离“因式分解”出来的变量替换，就能得到一个关于角变量 v 的可积方程。
10. Special Cases and Extensions | 特殊情况与扩展
Sometimes the equation is more naturally treated with the substitution x = vy (i.e., swapping roles) if the homogeneous function becomes simpler as a function of x/y. For example, if dy/dx = (x³ + y³)/(x y²), you might try v = x/y. The derivative transforms differently, but the method is analogous. In IB exams, however, y = vx is the standard approach and is usually sufficient.
有时，如果齐次函数化为 x/y 的函数更简单，可以自然地使用 x = vy 的代换（即角色互换）。例如，若 dy/dx = (x³ + y³)/(x y²)，你可以尝试 v = x/y。虽然求导转换有所不同，但方法是类似的。然而在IB考试中，y = vx 是标准方法，通常已经足够。
Equations of the form dy/dx = (ax + by + c)/(px + qy + r) where constants c and r are not zero can be reduced to homogeneous form by a translation of axes: set X = x – h, Y = y – k, choosing h and k to eliminate the constant terms. Then the substitution Y = V X becomes effective. This extension is occasionally seen in HL problem sets.
形如 dy/dx = (ax + by + c)/(px + qy + r) 且常数 c 和 r 不为零的方程，可以通过平移坐标轴化为齐次形式：设 X = x – h, Y = y – k，并选择 h 和 k 消去常数项。接着代换 Y = V X 就能生效。这种扩展在HL习题中偶有出现。
11. IB Exam Tips and Marking Points | IB 考试技巧与得分点
- Show the homogeneity check: Even if it seems obvious, a short line “Replace x by tx, y by ty – all t’s cancel, so homogeneous” earns method marks.
展示齐次性检验：即使看起来很明显，写一句“将 x 替换为 tx，y 替换为 ty——所有 t 消去，因此是齐次”就能获得方法分。 - State the substitution clearly: Let v = y/x, then y = vx, dy/dx = v + x dv/dx. This set-up is crucial and often carries marks for the correct derivative.
清晰陈述代换：令 v = y/x，则 y = vx，dy/dx = v + x dv/dx。这一设定至关重要，正确的导数通常有分值。 - Organize algebraic steps: After substitution, simplify to x dv/dx = … and show the separation step clearly. Messy algebra can lose accuracy marks even if the overall idea is right.
条理化代数步骤：代换后化简至 x dv/dx = …，并清楚展示分离变量步骤。混乱的代数容易丢掉准确度分数，即使整体思路正确。 - Integrate both sides with proper notation: Write the integral signs and include dx or dv. The right side integrates to ln|x| + C; remember the absolute value. If a definite integral is used for an initial condition, show the limits.
使用适当符号进行两边积分：写出积分号并带上 dx 或 dv。右侧积分为 ln|x| + C；记住加上绝对值。如果使用定积分处理初始条件，请标出积分限。 - General solution in simplest form: After back-substitution, aim for a clean expression, possibly with y² or an implicit relation. IB often accepts implicit forms unless stated otherwise.
最简形式的通解：回代后，力求表达清晰，可以是带 y² 的式子或隐式关系。除非题目另作要求，IB 通常接受隐式形式。 - Check for domain issues: If the solution contains ln or square roots, briefly mention x ≠ 0 or restrictions where the solution is valid. This shows deeper understanding.
检查定义域问题：如果解中含有 ln 或平方根，简要说明 x ≠ 0 或解的有效范围。这能展现更深入的理解。
12. Conclusion and Summary | 总结与概览
The substitution y = vx is a systematic technique for reducing homogeneous first-order differential equations to separable ones. By recognizing the zero-degree homogeneous structure, setting v = y/x, applying the product rule for differentiation, and carefully separating variables, you can tackle a wide variety of IB HL problems. The process reinforces core calculus skills: chain and product rules, integration with logarithmic and trigonometric functions, and algebraic manipulation. Practising with examples of increasing complexity will build fluency, and attention to details such as absolute values, constant renaming, and singular solutions will earn full marks. Remember, the key is to make the equation all about v and x, integrate, and then bring back y.
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