Solving Differential Equations: Methods and Applications | 微分方程的解法与应用

📚 Solving Differential Equations: Methods and Applications | 微分方程的解法与应用

Differential equations are among the most powerful tools in mathematics, used to model real-world phenomena ranging from population dynamics to electrical circuits. This article provides a concise review of the essential solution methods and their applications, tailored for A-level Further Mathematics and equivalent examinations.

微分方程是数学中最强大的工具之一，用于模拟从种群动态到电路的各种现实世界现象。本文为A-level进阶数学及同等考试精心总结了微分方程的核心解法及其应用。

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1. What is a Differential Equation? | 什么是微分方程？

A differential equation is an equation that relates a function to its derivatives. For example, the equation dy/dx = 3x² states that the derivative of y with respect to x equals three times x squared. The order of a differential equation is the highest derivative it contains.

微分方程是联系函数与其导数的方程。例如，方程 dy/dx = 3x² 表示 y 对 x 的导数等于 3x²。微分方程的阶是其中所含的最高导数阶数。

We classify equations as ordinary (involving ordinary derivatives) or partial (involving partial derivatives). In A-level mathematics, we focus on ordinary differential equations (ODEs). A solution may be given implicitly or explicitly, and generally contains arbitrary constants unless initial conditions are supplied.

我们将方程分为常微分方程（含普通导数）和偏微分方程（含偏导数）。在A-level数学中，我们重点研究常微分方程（ODE）。解可以隐式或显式给出，除非给定初始条件，否则一般含有任意常数。

F(x, y, dy/dx, d²y/dx², …) = 0

Understanding the terminology is crucial: a solution is a function y(x) that satisfies the equation; a general solution contains all possible solutions; a particular solution is obtained by fixing the arbitrary constants using extra conditions.

理解术语至关重要：解是满足方程的函数 y(x)；通解包含所有可能的解；特解是通过附加条件确定任意常数后得到的解。

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2. Separation of Variables | 可分离变量法

Separation of variables is used when a first-order differential equation can be written in the form dy/dx = f(x)g(y). We rearrange so that all y terms are on one side and all x terms are on the other, then integrate both sides.

当一阶微分方程可写成 dy/dx = f(x)g(y) 形式时，可使用可分离变量法。我们将其重新整理，使所有含 y 的项在一边，所有含 x 的项在另一边，然后对两边积分。

∫ (1/g(y)) dy = ∫ f(x) dx

For example, to solve dy/dx = x/y, multiply by y to get y dy = x dx. Integrating both sides gives ½y² = ½x² + C, or y² = x² + C.

例如，解 dy/dx = x/y，两边乘以 y 得 y dy = x dx。两边积分得 ½y² = ½x² + C，即 y² = x² + C。

For a more involved case, solve dy/dx = y² sin x with y(0) = 1. Separating gives dy/y² = sin x dx. Integrating yields -1/y = -cos x + C. Using y(0)=1 gives -1 = -1 + C, so C = 0. Hence y = 1/cos x, provided cos x ≠ 0.

再看一个稍复杂的例子：解 dy/dx = y² sin x 且 y(0) = 1。分离变量得 dy/y² = sin x dx。积分得 -1/y = -cos x + C。由 y(0)=1 得 -1 = -1 + C，所以 C = 0。因此 y = 1/cos x（要求 cos x ≠ 0）。

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3. First-Order Linear Differential Equations | 一阶线性微分方程

A first-order linear ODE has the form dy/dx + P(x)y = Q(x). To solve it, we use an integrating factor given by I = e^{∫P dx}. Multiplying both sides of the equation by I turns the left-hand side into the derivative of I·y.

一阶线性常微分方程具有形式 dy/dx + P(x)y = Q(x)。我们使用积分因子 I = e^{∫P dx} 来求解。将方程两边乘以 I 后，左边变为 I·y 的导数。

I·y = ∫ I·Q(x) dx + C

For instance, consider dy/dx + 2y = x. Here P(x)=2, so I = e^2x. Multiplying through and integrating gives e^2xy = ∫ x e^2x dx. Integrating by parts yields e^2xy = (x/2)e^2x - (1/4)e^2x + C, so y = x/2 - 1/4 + C e^-2x.

例如，考虑 dy/dx + 2y = x。这里 P(x)=2，所以 I = e^2x。两边相乘并积分得 e^2xy = ∫ x e^2x dx。分部积分得 e^2xy = (x/2)e^2x - (1/4)e^2x + C，因此 y = x/2 - 1/4 + C e^-2x。

Always remember to rewrite the equation in standard form before identifying P(x) and Q(x). If the equation is given as dy/dx = f(x) - P(x)y, rearrange it first.

在识别 P(x) 和 Q(x) 之前，务必先将方程改写为标准形式。如果方程以 dy/dx = f(x) - P(x)y 的形式给出，要先整理。

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4. Homogeneous Second-Order Linear Equations | 二阶常系数齐次线性微分方程

For an equation of the form a d²y/dx² + b dy/dx + c y = 0, we assume a solution of the form y = e^mx. Substituting gives the auxiliary equation am² + bm + c = 0.

对于形如 a d²y/dx² + b dy/dx + c y = 0 的方程，我们假设解为 y = e^mx。代入后得到辅助方程 am² + bm + c = 0。

am² + bm + c = 0

The nature of the roots, determined by the discriminant Δ = b² - 4ac, decides the general solution:

根的性质由判别式 Δ = b² - 4ac 决定，并据此写出通解：

• If Δ > 0 (distinct real roots m₁ and m₂): y = A e^m₁x + B e^m₂x.

  若 Δ > 0（两个不同实根 m₁ 和 m₂）：y = A e^m₁x + B e^m₂x。

• If Δ = 0 (repeated root m): y = (A + Bx) e^mx.

  若 Δ = 0（重根 m）：y = (A + Bx) e^mx。

• If Δ < 0 (complex roots m = α ± iβ): y = e^αx(A cos βx + B sin βx).

  若 Δ < 0（复数根 m = α ± iβ）：y = e^αx(A cos βx + B sin βx)。

For example, the equation y″ - 6y′ + 13y = 0 has auxiliary equation m² - 6m + 13 = 0, giving m = 3 ± 2i. Hence y = e^3x(A cos 2x + B sin 2x).

例如，方程 y″ - 6y′ + 13y = 0 的辅助方程为 m² - 6m + 13 = 0，解得 m = 3 ± 2i。因此 y = e^3x(A cos 2x + B sin 2x)。

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5. Finding Particular Solutions | 求特解

When the differential equation is non-homogeneous, such as a d²y/dx² + b dy/dx + c y = f(x), the general solution is the sum of the complementary function (solution of the homogeneous equation) and a particular integral.

当微分方程是非齐次的，例如 a d²y/dx² + b dy/dx + c y = f(x)，其通解为互补函数（齐次方程的解）与特解之和。

We find the particular integral by undetermined coefficients, selecting a trial solution based on f(x):

我们用待定系数法求特解，根据 f(x) 的形式选取试探解：

• If f(x) is a polynomial of degree n, try a polynomial of degree n.

  若 f(x) 是 n 次多项式，尝试一个 n 次多项式。

• If f(x) = k e^αx, try C e^αx, unless e^αx already appears in the complementary function.

  若 f(x) = k e^αx，尝试 C e^αx，除非 e^αx 已出现在互补函数中。

• If f(x) = k cos βx or k sin βx, try C cos βx + D sin βx.

  若 f(x) = k cos βx 或 k sin βx，尝试 C cos βx + D sin βx。

When the trial solution overlaps with the complementary function,

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