Second Order Linear Differential Equations with Variable Coefficients | 变系数二阶线性微分方程

📚 Second Order Linear Differential Equations with Variable Coefficients | 变系数二阶线性微分方程

In AQA A-Level Further Mathematics, second order linear differential equations can appear with coefficients that are functions of x rather than constants. This article covers the main methods for such equations: reduction of order when one solution is known, and the Euler-Cauchy equation, which can be transformed into a constant-coefficient problem.

在 AQA A-Level 进阶数学中，二阶线性微分方程的系数可以是 x 的函数，而不一定是常数。本文介绍这类方程的主要解法：已知一个解时的降阶法，以及可以转化为常系数方程的 Euler-Cauchy 方程。

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1. Overview and General Form | 概述与一般形式

A second order linear differential equation with variable coefficients has the general form

a(x)y″ + b(x)y′ + c(x)y = f(x),

where a(x) is not zero on the interval of interest, and a, b, c, f are functions of x. If f(x) = 0, the equation is homogeneous; otherwise it is non-homogeneous. Because the coefficients change with x, the usual auxiliary equation for constant coefficients does not apply directly.

变系数二阶线性微分方程的一般形式为

a(x)y″ + b(x)y′ + c(x)y = f(x),

其中在所讨论的区间上 a(x) ≠ 0，且 a、b、c、f 都是 x 的函数。若 f(x) = 0，方程为齐次方程；否则为非齐次方程。由于系数随 x 变化，常系数方程中常用的辅助方程法不能直接套用。

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2. Linearity and Superposition | 线性性与叠加原理

Consider the homogeneous equation

L[y] = a(x)y″ + b(x)y′ + c(x)y = 0.

The operator L is linear. This means that if y₁ and y₂ are two solutions of the homogeneous equation, then any linear combination A y₁ + B y₂ is also a solution, where A and B are arbitrary constants. Therefore the general solution requires two linearly independent solutions.

考虑齐次方程

L[y] = a(x)y″ + b(x)y′ + c(x)y = 0。

算子 L 是线性的。这意味着如果 y₁ 和 y₂ 是齐次方程的两个解，那么任意线性组合 A y₁ + B y₂ 也是解，其中 A 和 B 是任意常数。因此，通解需要两个线性无关的解。

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3. Normalised Form and the Wronskian | 标准形式与 Wronskian 行列式

Divide the homogeneous equation through by a(x) to write it in normalised form:

y″ + P(x)y′ + Q(x)y = 0.

If y₁ and y₂ are two solutions, their Wronskian is

W = y₁y₂′ − y₁′y₂.

The Wronskian satisfies the first order equation

W′ + P(x)W = 0.

This shows that W is either identically zero or never zero on an interval where P is continuous. A nonzero Wronskian confirms that the two solutions are linearly independent.

将齐次方程两边除以 a(x)，可写成标准形式：

y″ + P(x)y′ + Q(x)y = 0。

若 y₁ 和 y₂ 是两个解，它们的 Wronskian 行列式为

W = y₁y₂′ − y₁′y₂。

该 Wronskian 满足一阶方程

W′ + P(x)W = 0。

这表明在 P 连续的区间上，W 要么恒为零，要么处处不为零。Wronskian 不为零即可确认两个解线性无关。

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4. Reduction of Order | 降阶法

If one solution y₁ of the normalised equation y″ + P(x)y′ + Q(x)y = 0 is known, a second linearly independent solution can be found by setting

y₂ = u(x)y₁.

Substitute this into the differential equation. Since y₁ already satisfies the equation, the terms involving u cancel and you obtain

u″y₁ + u′(2y₁′ + P y₁) = 0.

Let v = u′. This becomes a first order separable equation in v:

v′/v = −(2y₁′/y₁ + P).

Integrating twice gives the reduction of order formula

u = ∫ [ e^(−∫P dx) / y₁² ] dx.

Then the general solution is y = A y₁ + B u y₁, where A and B are arbitrary constants.

如果已知标准形式方程 y″ + P(x)y′ + Q(x)y = 0 的一个解 y₁，可以通过设

y₂ = u(x)y₁

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