Formation of Second Order Differential Equations | 二阶微分方程的建立

📚 Formation of Second Order Differential Equations | 二阶微分方程的建立

In A-Level Mathematics, differential equations are used to model dynamic systems. A second order differential equation contains a second derivative and arises naturally in physics, engineering and geometry. This article focuses on the formation of such equations — i.e., how to translate a physical law or property into a mathematical equation involving d²y/dx² or d²x/dt².

在 A-Level 数学中，微分方程用于对动态系统建模。二阶微分方程包含二阶导数，并自然出现在物理、工程和几何中。本文聚焦于这类方程的建立——即如何将物理定律或性质转化为含有 d²y/dx² 或 d²x/dt² 的数学方程。

1. What is a Second Order Differential Equation? | 什么是二阶微分方程？

A second order differential equation is an equation that involves the second derivative of an unknown function. In general, it can be written in the form:

a(x) d²y/dx² + b(x) dy/dx + c(x) y = f(x)

If the function and its derivatives appear only to the first power and are not multiplied together, the equation is said to be linear. The order of the equation is determined by the highest derivative present. In the equation above, the presence of d²y/dx² makes it second order.

二阶微分方程是包含未知函数二阶导数的方程。一般可写成以下形式：

a(x) d²y/dx² + b(x) dy/dx + c(x) y = f(x)

如果函数及其导数只以一次幂出现且彼此不相乘，则称该方程为线性方程。方程的阶数由所含的最高阶导数决定。在上述方程中，由于含有 d²y/dx²，因此它是二阶的。

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2. Standard Notation and Terminology | 标准形式与记号

Different fields use different notations for derivatives. For a function y = y(x), the second derivative can be written as ‘d²y/dx²’ or as y′. When the independent variable is time, we often use dots: x′ for velocity and x″ for acceleration. The second derivative may also be written as y″ (read as ‘y double prime’).

不同领域使用不同的导数记号。对于函数 y = y(x)，二阶导数可以写成 d²y/dx² 或 y″。当自变量为时间时，常用点标记：x′ 表示速度，x″ 表示加速度。二阶导数也可以写成 y″（读作“y 两撇”）。

For a second order differential equation, the standard linear form is often written with time as the independent variable:

a(t) d²x/dt² + b(t) dx/dt + c(t) x = f(t)

When f(t) = 0, the equation is called homogeneous; otherwise it is non-homogeneous. When forming such equations, it is important to rearrange them into this standard form so that solving methods can be applied later.

对于二阶微分方程，标准线性形式通常以时间为自变量写成：

a(t) d²x/dt² + b(t) dx/dt + c(t) x = f(t)

当 f(t) = 0 时，方程称为齐次方程；否则称为非齐次方程。在建立这类方程时，需要将其整理成标准形式，以便后续应用求解方法。

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3. Formation from Newton’s Second Law | 由牛顿第二定律建立

Newton’s second law states that the net force acting on an object equals its mass times its acceleration, F = ma. Since acceleration is the second derivative of displacement with respect to time, we immediately obtain a second order differential equation when the force is expressed in terms of displacement, velocity or time.

牛顿第二定律指出：物体所受合力等于其质量乘以加速度，即 F = ma。由于加速度是位移对时间的二阶导数，因此只要合力用位移、速度或时间表示，就能立即得到一个二阶微分方程。

Consider a mass m attached to a spring. The spring exerts a restoring force −kx, where x is the displacement from equilibrium and k is the spring constant. If a damping force proportional to velocity, −c dx/dt, and an external force F(t) are also present, then:

m d²x/dt² = −c dx/dt − kx + F(t)

Rearranging the terms gives the standard form:

m d²x/dt² + c dx/dt + kx = F(t)

This is a second order linear differential equation with constant coefficients. The sign of each term is crucial: the restoring force is opposite to the displacement, and the damping force is opposite to the velocity, so each appears with a negative sign on the right-hand side.

设想一个质量为 m 的物体连接在弹簧上。弹簧施加恢复力 −kx，其中 x 是偏离平衡位置的位移，k 是劲度系数。如果还存在与速度成正比的阻尼力 −c dx/dt 以及外力 F(t)，则有：

m d²x/dt² = −c dx/dt − kx + F(t)

移项整理后得到标准形式：

m d²x/dt² + c dx/dt + kx = F(t)

这是一个常系数二阶线性微分方程。各项的符号至关重要：恢复力与位移方向相反，阻尼力与速度方向相反，因此它们在等式右侧都带负号。

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4. Formation from Simple Harmonic Motion | 由简谐运动建立

Simple harmonic motion (SHM) is defined by the property that acceleration is proportional to displacement but directed towards the equilibrium position. This can be written as a = −ω²x, where ω is the angular frequency. Replacing acceleration by d²x/dt² gives:

d²x/dt² = −ω²x

Bringing all terms to one side produces the classic SHM differential equation:

d²x/dt² + ω²x = 0

For example, a particle moves with SHM and has period T. Since T = 2π/ω, the angular frequency is ω = 2π/T. Substituting this value forms the differential equation for the motion:

d²x/dt² + (2π/T)² x = 0

The formation relies on recognizing the physical relationship between acceleration and displacement. The negative sign is essential: it indicates that the acceleration always points back towards the equilibrium position.

简谐运动（SHM）的定义是：加速度与位移成正比，但方向总是朝向平衡位置。可写为 a = −ω²x，其中 ω 为角频率。将加速度替换为 d²x/dt²，可得：

d²x/dt² = −ω²x

将所有项移到一侧，就得到经典的简谐运动微分方程：

d²x/dt² + ω²x = 0

例如，一个质点做简谐运动，周期为 T。由于 T = 2π/ω，角频率 ω = 2π/T。代入后即可建立运动微分方程：

d²x/dt² + (2π/T)² x = 0

这种建立方法依赖于识别加速度与位移之间的物理关系。负号至关重要：它表示加速度总是指向平衡位置。

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5. Formation from Electrical Circuits | 由电路建立

Series electrical circuits often lead to second order differential equations. In an LCR circuit containing an inductor, a resistor and a capacitor, the sum of voltage drops around the loop equals the applied voltage E(t). The voltage across an inductor is L di/dt, across a resistor is Ri, and across a capacitor is q/C, where i = dq/dt is the current and q is the charge.

串联电路常常会产生二阶微分方程。在包含电感、电阻和电容的 LCR 电路中，回路中的电压降之和等于外加电压 E(t)。电感两端电压为 L di/dt，电阻两端电压为 Ri，电容两端电压为 q/C，其中 i = dq/dt 为电流，q 为电荷。

Applying Kirchhoff’s voltage law gives:

L di/dt + Ri + q/C = E(t)

Since i = dq/dt, we also have di/dt = d²q/dt². Therefore:

L d²q/dt² + R dq/dt + q/C = E(t)

This is the second order differential equation that governs the charge in the circuit. If the applied voltage is zero, the circuit is homogeneous and represents free oscillations.

应用基尔霍夫电压定律得：

L di/dt + Ri + q/C = E(t)

由于 i = dq/dt，因此 di/dt = d²q/dt²。于是：

L d²q/dt² + R dq/dt + q/C = E(t)

这就是控制电路中电荷变化的二阶微分方程。如果外加电压为零，电路是齐次的，表示自由振荡。

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6. Formation by Eliminating Arbitrary Constants | 通过消去任意常数建立

When a general solution is given, we can form a differential equation by differentiating it enough times and eliminating the arbitrary constants. This is a purely algebraic method often used to verify that a given family of curves satisfies a particular differential equation.

当给出通解时，我们可以通过多次求导并消去任意常数来建立微分方程。这是一种纯代数方法，常用于验证给定的曲线族满足某一特定微分方程。

Consider the family of curves y = A cos 3x + B sin 3x. Differentiating twice gives:

dy/dx = −3A sin 3x + 3B cos 3x

d²y/dx² = −9A cos 3x − 9B sin 3x = −9y

Thus we form the second order equation:

d²y/dx² + 9y = 0

In this process, the two arbitrary constants A and B disappeared automatically because both y and its second derivative were expressed as the same combination of sin 3x and cos 3x.

考虑曲线族 y = A cos 3x + B sin 3x。求导两次得：

dy/dx = −3A sin 3x + 3B cos 3x

d²y/dx² = −9A cos 3x − 9B sin 3x = −9y

因此得到二阶方程：

d²y/dx² + 9y = 0

在这个过程中，任意常数 A 和 B 自动消失，因为 y 与其二阶导数都表达为 sin 3x 和 cos 3x 的同一组合。

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7. Formation from Geometric Properties | 由几何性质建立

Some curves are defined by conditions involving their slopes or rates of change of slope. The slope of a curve is dy/dx, and the rate of change of the slope is the second derivative d²y/dx². If a problem states a relationship between these quantities and the coordinates, we can write down a second order differential equation directly.

有些曲线由涉及斜率或斜率变化率的条件定义。曲线的斜率是 dy/dx，斜率的变化率是二阶导数 d²y/dx²。如果题目给出这些量与坐标之间的某个关系，我们就可以直接写出二阶微分方程。

For example, suppose a curve has the property that the rate of change of its slope is always equal to the y-coordinate. Then:

d²y/dx² = y

Writing this in standard form gives:

d²y/dx² − y = 0

Similarly, if the slope’s rate of change is the negative of the x-coordinate, we obtain d²y/dx² = −x. This type of formation is common in curve-sketching contexts and simple mechanics problems.

例如，假设某曲线具有这样的性质：其斜率的变化率总是等于该点的 y 坐标。则：

d²y/dx² = y

写成标准形式即为：

d²y/dx² − y = 0

类似地，如果斜率变化率等于 x 坐标的相反数，则得到 d²y/dx² = −x。这种建立在曲线作图背景和简单力学问题中很常见。

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8. Initial and Boundary Conditions | 初始条件与边界条件

Forming a second order differential equation is only the first step. To determine a unique solution, we need conditions that allow us to find the arbitrary constants appearing in the general solution. Since a second order equation has two arbitrary constants, we require two conditions.

建立二阶微分方程只是第一步。为了确定唯一解，需要若干条件来求通解中的任意常数。由于二阶方程含有两个任意常数，因此需要两个条件。

Initial conditions specify the value of the dependent variable and its first derivative at a single point, usually t = 0. For example, for a particle’s displacement x(t), we might be told:

x(0) = 2, x′(0) = 0

Boundary conditions specify values at two different points, such as y(0) = 0 and y(1) = 0. When forming a differential equation from a physical problem, these conditions are often given as part of the initial description. Always write them down clearly.

初始条件指定了因变量及其一阶导数在同一个点（通常为 t = 0）的值。例如，对于质点位移 x(t)，可能给出：

x(0) = 2, x′(0) = 0

边界条件则指定两个不同点处的值，例如 y(0) = 0 和 y(1) = 0。从物理问题建立微分方程时，这些条件通常包含在初始描述中。务必将其明确写出。

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9. Worked Example: Forming from a Physical Scenario | 实例：从物理情境建立

Let us combine the ideas from the previous sections in a worked example. A 2 kg object moves in a straight line. The net force acting on it is given by F = −8x − 4v, where x is the displacement from equilibrium in metres and v is the velocity in m/s. Form the differential equation for x(t).

让我们用一个实例综合前面的内容。一个 2 kg 的物体在直线上运动。它所受的合力为 F

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