📚 Forced Linear Differential Equations | IB数学：受迫线性微分方程
A forced linear differential equation is a linear ordinary differential equation in which a non-zero external function drives the system. In IB Mathematics Analysis and Approaches HL, these equations appear in the calculus option, often as second-order equations with constant coefficients. They model real situations where an outside influence, such as a periodic force, a voltage input, or a gravitational pull, acts on a system.
受迫线性微分方程是指由非零外部函数驱动系统的线性常微分方程。在 IB 数学分析与方法 HL 中，这类方程出现在微积分选修部分，通常为常系数二阶方程。它们用于建模外部影响（如周期力、电压输入或引力）作用于系统时的真实情景。
1. What is a Forced Linear Differential Equation? | 什么是受迫线性微分方程？
A general linear differential equation of order n has the form:
aₙ y⁽ⁿ⁾ + aₙ₋₁ y⁽ⁿ⁻¹⁾ + … + a₁ y′ + a₀ y = f(x)
When f(x) is identically zero, the equation is called homogeneous. When f(x) is not zero, the equation is called non-homogeneous, or forced. The function f(x) is the forcing term, which represents an external input that drives the system independently of its current state.
一般 n 阶线性微分方程的形式为：
aₙ y⁽ⁿ⁾ + aₙ₋₁ y⁽ⁿ⁻¹⁾ + … + a₁ y′ + a₀ y = f(x)
当 f(x) 恒为零时，方程称为齐次方程；当 f(x) 不为零时，方程称为非齐次方程或受迫方程。函数 f(x) 称为强迫项，它表示一个外部输入，独立于系统当前状态而驱动系统。
2. General Solution: Complementary Function + Particular Integral | 通解：补函数 + 特解
For a linear non-homogeneous differential equation, the general solution is the sum of two parts:
y(x) = y_c(x) + y_p(x)
The complementary function y_c solves the associated homogeneous equation (with f replaced by zero). The particular integral y_p is any single solution of the full non-homogeneous equation. The constant coefficients in y_c are determined by initial or boundary conditions.
对于线性非齐次微分方程，通解由两部分组成：
y(x) = y_c(x) + y_p(x)
补函数 y_c 是对应齐次方程（令 f 为零）的解；特解 y_p 是非齐次方程的任意一个解。y_c 中的任意常数由初始条件或边界条件确定。
3. The Auxiliary Equation | 辅助方程
To find the complementary function for a constant-coefficient second-order equation such as:
ay″ + by′ + cy = 0
we substitute y = eʳˣ. This gives the auxiliary equation:
ar² + br + c = 0
If the roots are real and distinct r₁ and r₂, then y_c = Aeʳ¹ˣ + Beʳ²ˣ. If the roots are equal r, then y_c = (A + Bx)eʳˣ. If the roots are complex α ± βi, then y_c = eᵅˣ(A sin βx + B cos βx).
要求常系数二阶方程的补函数，例如：
ay″ + by′ + cy = 0
我们令 y = eʳˣ，得到辅助方程：
ar² + br + c = 0
若根为相异实根 r₁ 和 r₂，则 y_c = Aeʳ¹ˣ + Beʳ²ˣ；若根为重根 r，则 y_c = (A + Bx)eʳˣ；若根为共轭复数 α ± βi，则 y_c = eᵅˣ(A sin βx + B cos βx)。
4. Methods for Finding the Particular Integral | 求特解的方法
There are several standard methods for finding a particular integral. In IB, the most commonly tested method is the method of undetermined coefficients. This works when the forcing term f(x) belongs to a limited class of functions: exponentials, polynomials, sine and cosine, or products and sums of these.
求特解的常用方法有多种。在 IB 考试中，最常考查的方法是待定系数法。它适用于强迫项 f(x) 属于较有限函数类别的情形：指数函数、多项式、正弦和余弦函数，或这些函数的乘积与和。
5. Undetermined Coefficients: Basic Rules | 待定系数法：基本规则
The table below summarizes the trial forms for common forcing terms. If the trial function is already a solution of the homogeneous equation, multiply the trial by x (or by x² in the case of resonance).
下表总结了常见强迫项所对应的试验函数。如果该试验函数已经是齐次方程的解，则将试验函数乘以 x（共振情形下乘以 x²）。
| Forcing term f(x) | Trial form yₚ(x) |
| k eᵃˣ | C eᵃˣ |
| Polynomial of degree n | Aₙxⁿ + Aₙ₋₁xⁿ⁻¹ + … + A₀ |
| k sin(bx) or k cos(bx) | C sin(bx) + D cos(bx) |
| eᵃˣ sin(bx) or eᵃˣ cos(bx) | eᵃˣ(C sin(bx) + D cos(bx)) |
After substituting yₚ into the differential equation, compare coefficients to determine the unknown constants. This method is efficient because the derivatives of these trial functions have forms similar to the trial itself.
将 yₚ 代入微分方程后，比较系数即可确定未知常数。此方法十分高效，因为这些试验函数的导数形式与试验函数本身相似。
6. Worked Example: Exponential Forcing | 例题：指数型强迫项
Solve the equation:
y″ − 3y′ + 2y = 10e⁻ˣ
First find the complementary function. The auxiliary equation is r² − 3r + 2 = 0, so (r − 1)(r − 2) = 0, giving r = 1 and r = 2. Therefore y_c = Aeˣ + Be²ˣ.
首先求补函数。辅助方程为 r² − 3r + 2 = 0，即 (r − 1)(r − 2) = 0，得 r = 1 和 r = 2。因此 y_c = Aeˣ + Be²ˣ。
For the particular integral, try yₚ = Ce⁻ˣ. Then yₚ′ = −Ce⁻ˣ and yₚ″ = Ce⁻ˣ. Substituting:
Ce⁻ˣ − 3(−Ce⁻ˣ) + 2(Ce⁻ˣ) = 10e⁻ˣ
This simplifies to (C + 3C + 2C)e⁻ˣ = 6Ce⁻ˣ = 10e⁻ˣ, so C = 5/3. The general solution is:
y = Aeˣ + Be²ˣ + (5/3)e⁻ˣ
对于特解，试验 yₚ = Ce⁻ˣ。则 yₚ′ = −Ce⁻ˣ，yₚ″ = Ce⁻ˣ。代入得：
Ce⁻ˣ − 3(−Ce⁻ˣ) + 2(Ce⁻ˣ) = 10e⁻ˣ
化简为 (C + 3C + 2C)e⁻ˣ = 6Ce⁻ˣ = 10e⁻ˣ，所以 C = 5/3。通解为：
y = Aeˣ + Be²ˣ + (5/3)e⁻ˣ
7. Forced Damped Oscillators | 受迫阻尼振动
Many physical systems are described by the forced damped oscillator equation:
m y″ + c y′ + k y = F₀ cos(ωt)
Here m is mass, c is the damping coefficient, k is the spring constant, and F₀ cos(ωt) is the external periodic force. The complementary function represents the transient motion, which decays over time when damping is present. The particular integral represents the steady-state response, which oscillates with the same frequency as the forcing term.
许多物理系统可用受迫阻尼振动方程描述：
m y″ + c y′ + k y = F₀ cos(ωt)
其中 m 为质量，c 为阻尼系数，k 为弹簧常数，F₀ cos(ωt) 为外部周期力。补函数表示瞬态运动，在存在阻尼时随时间衰减；特解表示稳态响应，其振荡频率与强迫项相同。
8. Resonance | 共振
Resonance occurs when the forcing frequency matches the natural frequency of the system. In the undamped case, the trial function for a sinusoidal forcing term is multiplied by x, producing a particular solution whose amplitude grows without bound as t increases.
当强迫频率与系统固有频率相等时会发生共振。在无阻尼情形下，正弦型强迫项的试验函数会乘以 x，从而产生振幅随 t 增大而无界增长的特解。
For example, consider the equation:
y″ + 4y = cos(2x)
The complementary function is y_c = A sin 2x + B cos 2x. Since the forcing term cos(2x) is already part of y_c, the trial must be yₚ = x(C sin 2x + D cos 2x). Substitution yields a particular solution of the form yₚ = (x/4) sin 2x, whose amplitude grows linearly.
例如，考虑方程：
y″ + 4y = cos(2x)
补函数为 y_c = A sin 2x + B cos 2x。由于强迫项 cos(2x) 已经是 y_c 的一部分，试验函数必须取 yₚ = x(C sin 2x + D cos 2x)。代入后可得特解 yₚ = (x/4) sin 2x，其振幅随 x 线性增长。
9. Initial Value Problems | 初值问题
To solve an initial value problem, first write the full general solution y = y_c + yₚ. Then apply the initial conditions such as y(0) and y′(0) to determine the arbitrary constants A and B in y_c. These constants never appear in yₚ, because yₚ is uniquely determined up to the form of the trial function.
求解初值问题时，首先写出完整的通解 y = y_c + yₚ，然后将初始条件（如 y(0) 和 y′(0)）代入，以确定 y_c 中的任意常数 A 和 B。这些常数不会出现在 yₚ 中，因为特解的形式是唯一确定的。
For instance, for the equation y″ − 3y′ + 2y = 10e⁻ˣ with y(0) = 1 and y′(0) = 0, we use y = Aeˣ + Be²ˣ + (5/3)e⁻ˣ. Then y(0) = A + B + 5/3 = 1, and y′(0) = A + 2B − 5/3 = 0. Solving gives A and B, yielding the unique solution.
例如，对于方程 y″ − 3y′ + 2y = 10e⁻ˣ，且 y(0) = 1，y′(0) = 0，我们使用 y = Aeˣ + Be²ˣ + (5/3)e⁻ˣ。则 y(0) = A + B + 5/3 = 1，y′(0) = A + 2B − 5/3 = 0。解得 A 和 B，从而得到唯一解。
10. Transient and Steady-State Behaviour | 瞬态与稳态行为
For a forced system with damping, the complementary function contains real negative exponents or exponentially decaying oscillations, so it tends to zero as x → ∞. This part is called the transient response. The particular integral persists for all time and is called the steady-state response. In IB questions, you may be asked to identify the long-term behaviour without fully solving the initial value problem.
对于有阻尼的受迫系统，补函数包含负实指数或指数衰减振荡，因此当 x → ∞ 时趋于零。这部分称为瞬态响应。特解持续存在，称为稳态响应。在 IB 考题中，可能会要求你在不完全求解初值问题的情况下判断长期行为。
If the forcing term is periodic, the steady-state solution is periodic with the same frequency as the forcing term, though its phase may be shifted. If the forcing term is an exponential growing function, the steady-state solution also grows exponentially, and the transient part becomes negligible.
若强迫项是周期的，则稳态解也是周期的，且频率与强迫项相同，但可能存在相位移动。若强迫项是指数增长函数，则稳态解也指数增长，而瞬态部分可忽略不计。
11. Applications | 应用
Forced linear differential equations appear in many fields. In physics, they describe driven harmonic oscillators and electrical RLC circuits. In chemistry, they model coupled reaction rates under external concentration changes. In economics, they represent growth models with external investment shocks. The method of undetermined coefficients is directly applicable whenever the forcing function is a sum or product of exponentials, polynomials, and sine/cosine functions.
受迫线性微分方程在许多领域都有应用。在物理学中，它们描述受迫谐振子与 RLC 电路；在化学中，它们在外界浓度变化下模拟耦合反应速率；在经济学中，它们表示带有外部投资冲击的增长模型。当强迫函数为指数、多项式和正弦/余弦函数的和或积时，待定系数法可直接应用。
Understanding the structure of the solution — the addition of a complementary function and a particular integral — is essential for IB exam questions, where you must combine algebraic manipulation with calculus techniques.
理解解的结构——补函数与特解之和——是 IB 考试问题的关键，你必须将代数运算与微积分技巧结合起来。
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