Methods for Solving Ordinary Differential Equations | 常微分方程求解方法

📚 Methods for Solving Ordinary Differential Equations | 常微分方程求解方法

Ordinary differential equations (ODEs) form the backbone of mathematical modelling in physics, engineering, economics and biology. Mastering the standard solution techniques is essential for any advanced mathematics examination, whether you are studying A-Level Further Mathematics, IB HL, or equivalent curricula.

常微分方程是物理、工程、经济与生物领域中数学建模的基石。无论你正在学习A-Level进阶数学、IB高级课程还是其他同等体系，熟练掌握标准求解方法都是考试取得高分的关键。

1. First-Order Separable Equations | 一阶可分离变量方程

The simplest type of first-order ODE is the separable equation, which can be written in the form dy/dx = f(x)g(y). The solution strategy is to rearrange the equation so that all terms involving y are on one side and all terms involving x are on the other, then integrate both sides.

最简单的一阶常微分方程是可分离变量方程，其形式为 dy/dx = f(x)g(y)。求解策略是将方程重新排列，使含有 y 的项在一边，含有 x 的项在另一边，然后对两边分别积分。

∫ 1/g(y) dy = ∫ f(x) dx + C

For example, consider dy/dx = 2xy. Separating variables gives (1/y) dy = 2x dx, and integrating yields ln|y| = x² + C, hence y = Ce^(x²). Remember to include the constant of integration C, and be careful with absolute values when integrating 1/y.

例如，考虑 dy/dx = 2xy。分离变量得到 (1/y) dy = 2x dx，积分得 ln|y| = x² + C，因此 y = Ce^(x²)。注意务必写上积分常数 C，并且在积分 1/y 时要小心绝对值符号。

2. First-Order Linear Equations and Integrating Factors | 一阶线性方程与积分因子

A first-order linear ODE has the standard form dy/dx + P(x)y = Q(x). When the equation cannot be separated, we use an integrating factor to convert the left-hand side into an exact derivative.

一阶线性常微分方程的标准形式为 dy/dx + P(x)y = Q(x)。当方程无法分离变量时，我们使用积分因子将左边转化为一个精确的导数。

Integrating factor: I(x) = e^(∫ P(x) dx)

Multiplying both sides by I(x) transforms the equation into d/dx [I(x)y] = I(x)Q(x). Then, integrate both sides with respect to x and solve for y. For instance, dy/dx + 2y = e^x has P(x) = 2, so I(x) = e^(2x). The solution is y = (1/3)e^x + Ce^(−2x).

将方程两边乘以 I(x)，可将其转化为 d/dx [I(x)y] = I(x)Q(x)。然后对 x 积分并解出 y。例如，dy/dx + 2y = e^x 中 P(x) = 2，所以 I(x) = e^(2x)，解得 y = (1/3)e^x + Ce^(−2x)。

3. Exact Differential Equations | 恰当微分方程

An ODE of the form M(x,y)dx + N(x,y)dy = 0 is called exact if ∂M/∂y = ∂N/∂x. In this case, there exists a function F(x,y) such that dF = M dx + N dy, and the general solution is F(x,y) = C.

形如 M(x,y)dx + N(x,y)dy = 0 的常微分方程称为恰当方程，如果它满足 ∂M/∂y = ∂N/∂x。此时存在一个函数 F(x,y)，使得 dF = M dx + N dy，通解为 F(x,y) = C。

F(x,y) = ∫ M dx + ∫ (N − ∂/∂y ∫ M dx) dy = C

To find F, integrate M with respect to x (keeping y constant), then determine the “constant” of integration by using the equation ∂F/∂y = N. If the equation is not exact, you may sometimes find an integrating factor that depends only on x or only on y to make it exact.

求 F 的方法是：先对 x 积分 M（此时 y 视为常数），然后利用 ∂F/∂y = N 确定积分常数。如果方程不是恰当的，有时可以找到只依赖于 x 或只依赖于 y 的积分因子，使其变为恰当方程。

4. Second-Order Equations Reducible to First Order | 可降阶的二阶方程

There are two classic situations where a second-order ODE can be reduced to a first-order problem. The first is when the dependent variable y is absent, so we substitute p = dy/dx and reduce the equation to first order in p and x.

有两种经典情形可以将二阶方程降为一阶问题。第一种是因变量 y 不出现，此时令 p = dy/dx，将方程化为关于 p 和 x 的一阶方程。

The second situation is when the independent variable x is absent, so we substitute p = dy/dx and use the chain rule to write d²y/dx² = p(dp/dy). The resulting first-order equation is then solved in terms of p and y.

第二种情形是自变量 x 不出现，此时令 p = dy/dx，并利用链式法则将 d²y/dx² 写为 p(dp/dy)，得到关于 p 和 y 的一阶方程再行求解。

For example, the equation y(d²y/dx²) = (dy/dx)² is solved by the second method: writing p = dy/dx gives yp(dp/dy) = p², so dp/p = dy/y, leading to p = C₁y and finally y = C₂e^(C₁x).

例如，方程 y(d²y/dx²) = (dy/dx)² 可用第二种方法求解：令 p = dy/dx，得 yp(dp/dy) = p²，即 dp/p = dy/y，解得 p = C₁y，最终 y = C₂e^(C₁x)。

5. Homogeneous Linear Equations with Constant Coefficients | 常系数齐次线性方程

For an nth-order linear homogeneous equation with constant coefficients, aⁿy⁽ⁿ⁾ + aₙ₋₁y⁽ⁿ⁻¹⁾ + … + a₁y’ + a₀y = 0, we assume a solution of the form y = e^(rx). Substituting yields the auxiliary (characteristic) equation.

对于 n 阶常系数齐次线性方程 aⁿy⁽ⁿ⁾ + aₙ₋₁y⁽ⁿ⁻¹⁾ + … + a₁y’ + a₀y = 0，我们假设解的形式为 y = e^(rx)。代入后得到辅助方程（特征方程）。

aₙrⁿ + aₙ₋₁rⁿ⁻¹ + … + a₁r + a₀ = 0

For the second-order case a y” + b y’ + c y = 0, the auxiliary equation ar² + br + c = 0 has three cases:

对于二阶情形 ay” + by’ + cy = 0，辅助方程 ar² + br + c = 0 有三种情况：

Distinct real roots r₁ ≠ r₂: y = C₁e^(r₁x) + C₂e^(r₂x).

相异实根 r₁ ≠ r₂：y = C₁e^(r₁x) + C₂e^(r₂x)。
Repeated real root r₀: y = (C₁ + C₂x)e^(r₀x).

重实根 r₀：y = (C₁ + C₂x)e^(r₀x)。
Complex conjugate roots α ± βi: y = e^(αx)(C₁cos βx + C₂sin βx).

共轭复根 α ± βi：y = e^(αx)(C₁cos βx + C₂sin βx)。

For higher-order equations, add extra factors of x for each repeated root and multiply by sine and cosine terms for complex pairs, extending the same logic.

对于高阶方程，每个重根需要额外乘上 x 的幂次因子，复根对则乘以正弦和余弦项，逻辑完全相同。

6. Non-Homogeneous Equations: Method of Undetermined Coefficients | 非齐次方程：待定系数法

For the non-homogeneous equation ay” + by’ + cy = f(x), the general solution is the sum of the complementary function (solution to the homogeneous equation) and a particular integral: y = y_c + y_p.

对于非齐次方程 ay” + by’ + cy = f(x)，通解等于补函数（齐次方程的解）与特解之和：y = y_c + y_p。

The method of undetermined coefficients applies when f(x) is a polynomial, exponential, sine, cosine, or a product/sum of these. The form of the particular integral is guessed based on f(x):

待定系数法适用于 f(x) 为多项式、指数函数、正弦、余弦或它们的和、积的情形。特解的形式根据 f(x) 来猜测：

f(x) = Pₙ(x) (polynomial): try y_p = x^k(aₙxⁿ + … + a₁x + a₀).

f(x) = Pₙ(x)（多项式）：尝试 y_p = x^k(aₙxⁿ + … + a₁x + a₀)。
f(x) = Ae^(αx): try y_p = Bx^k e^(αx).

f(x) = Ae^(αx)：尝试 y_p = Bx^k e^(αx)。
f(x) = A sin βx or A cos βx: try y_p = x^k(B sin βx + C cos βx).

f(x) = A sin βx 或 A cos βx：尝试 y_p = x^k(B sin βx + C cos βx)。

Here, k is the smallest non-negative integer that ensures no term of y_p is a solution of the homogeneous equation. This is known as the “resonance” or “overlap” rule. For example, y” − 3y’ + 2y = e^x: the homogeneous solution has e^x, so multiply by x: try y_p = Ax e^x.

其中 k 是使 y_p 中没有任何一项是齐次方程解的最小非负整数，这被称为”共振”或”重叠”法则。例如 y” − 3y’ + 2y = e^x 中，齐次解含有 e^x，所以特解需要乘以 x：尝试 y_p = Ax e^x。

7. Variation of Parameters | 参数变易法

When the forcing function f(x) is not of the simple forms suitable for undetermined coefficients, or when the ODE has variable coefficients, we use variation of parameters. For a second-order equation y” + P(x)y’ + Q(x)y = f(x), we first find two linearly independent solutions y₁ and y₂ of the homogeneous equation.

当非齐次项 f(x) 不属于适合待定系数法的简单形式，或方程具有变系数时，我们使用参数变易法。对于二阶方程 y” + P(x)y’ + Q(x)y = f(x)，首先求出齐次方程的两个线性无关解 y₁ 和 y₂。

y_p = u₁y₁ + u₂y₂, where u₁’ = −y₂f / W, u₂’ = y₁f / W

where W = y₁y₂’ − y₂y₁’ is the Wronskian. Integrating u₁’ and u₂’ gives u₁ and u₂, yielding a particular solution. This method is more general but requires solving integrals that may be challenging.

其中 W = y₁y₂’ − y₂y₁’ 为朗斯基行列式。对 u₁’ 和 u₂’ 积分得到 u₁ 和 u₂，从而得到特解。该方法适用范围更广，但需要计算可能较为复杂的积分。

8. The Euler–Cauchy Equation | 欧拉–柯西方程

The Euler–Cauchy (or equidimensional) equation has the form x²y” + bxy’ + cy = 0 for x > 0. Its distinguishing feature is that each derivative is multiplied by a power of x matching its order.

欧拉–柯西方程（等维方程）的形式为 x²y” + bxy’ + cy = 0（x > 0）。其显著特征是每一项的导数阶数恰好与 x 的幂次相匹配。

We solve it by making the substitution x = eᵗ, or equivalently by trying y = xᵐ. Substituting y = xᵐ yields the auxiliary equation:

我们通过代换 x = eᵗ 来求解，或者等价地尝试 y = xᵐ。代入 y = xᵐ 后得到辅助方程：

m(m−1) + bm + c = 0, i.e. m² + (b−1)m + c = 0

The three cases mirror the constant-coefficient situation: distinct real roots m₁ ≠ m₂ give y = C₁x^(m₁) + C₂x^(m₂); a repeated root m₀ gives y = (C₁ + C₂ ln x)x^(m₀); and complex roots m = α ± βi give y = x^α[C₁cos(β ln x) + C₂sin(β ln x)].

三种情况与常系数情形完全对应：相异实根 m₁ ≠ m₂ 给出 y = C₁x^(m₁) + C₂x^(m₂)；重根 m₀ 给出 y = (C₁ + C₂ ln x)x^(m₀)；复根 m = α ± βi 给出 y = x^α[C₁cos(β ln x) + C₂sin(β ln x)]。

9. Power Series Solutions | 幂级数解法

When an ODE has variable coefficients that do not fit the Euler–Cauchy form, we turn to power series methods. We assume a solution of the form y = Σₙ₌₀^∞ aₙ(x − x₀)ⁿ and substitute it into the differential equation to obtain a recurrence relation for the coefficients aₙ.

当常微分方程具有不适合欧拉–柯西形式的变系数时，我们采用幂级数解法。假设解的形式为 y = Σₙ₌₀^∞ aₙ(x − x₀)ⁿ，代入微分方程后得到系数 aₙ 的递推关系。

For example, in solving y” − 2xy’ + y = 0 about x₀ = 0, we substitute the series, shift indices so all sums have the same power of x, and equate coefficients. The recurrence relation determines every aₙ in terms of a₀ and a₁, producing two linearly independent series solutions.

例如，在求 y” − 2xy’ + y = 0 关于 x₀ = 0 的幂级数解时，我们代入级数、调整下标使所有求和项的 x 幂次一致，然后令各项系数相等。递推关系用 a₀ 和 a₁ 表示所有 aₙ，从而得到两个线性无关的级数解。

If x₀ is a regular singular point, the Frobenius method with y = xʳΣₙ₌₀^∞ aₙxⁿ is required. The indicial equation determines the possible values of r, and standard theory distinguishes three cases based on whether the roots of the indicial equation differ by an integer.

如果 x₀ 是正则奇点，则需要采用 Frobenius 方法，设 y = xʳΣₙ₌₀^∞ aₙxⁿ。指标方程确定 r 的可能取值，标准理论根据指标方程两根之差是否为整数分成三种情形。

10. Systems of First-Order Linear ODEs | 一阶线性微分方程组

A system of first-order linear ODEs can be written in matrix form as dy/dx = Ay, where y is a vector of unknown functions and A is a constant matrix. The solution is obtained by diagonalising A: if A has eigenvectors v₁, v₂ with eigenvalues λ₁, λ₂, then the general solution is a linear combination of vᵢe^(λᵢx).

一阶线性常微分方程组可以写成矩阵形式 dy/dx = Ay，其中 y 是未知函数向量，A 是常数矩阵。求解的关键是将 A 对角化：若 A 的特征向量为 v₁、v₂，对应的特征值为 λ₁、λ₂，则通解是 vᵢe^(λᵢx) 的线性组合。

y = C₁v₁e^(λ₁x) + C₂v₂e^(λ₂x)

For complex eigenvalues, the real and imaginary parts of the complex solutions provide two real solutions. For repeated eigenvalues, we may need generalised eigenvectors, giving extra factors of x in the solution. This matrix method elegantly connects linear algebra with differential equations.

对于复特征值，复解的实部和虚部给出两个实解。对于重特征值，可能需要广义特征向量，此时解中会出现 x 的额外因子。矩阵方法将线性代数与微分方程优美地联系起来。

11. Modelling with Differential Equations | 微分方程建模应用

Differential equations arise naturally in modelling real-world phenomena. In population dynamics, the logistic equation dP/dt = kP(1 − P/K) describes growth limited by carrying capacity K. In physics, Newton’s law of cooling states that the rate of temperature change is proportional to the temperature difference between the object and its surroundings.

微分方程在现实世界建模中无处不在。在种群动力学中，逻辑斯蒂方程 dP/dt = kP(1 − P/K) 描述了受环境容纳量 K 限制的增长。在物理学中，牛顿冷却定律表明温度变化率与物体和环境之间的温差成正比。

In electrical circuits, Kirchhoff’s voltage law applied to an RL circuit gives L(di/dt) + Ri = E(t), which is a first-order linear ODE solved by an integrating factor. In mechanics, Hooke’s law combined with Newton’s second law yields the damped harmonic oscillator equation mx” + cx’ + kx = 0, whose solution behaviour depends on whether the damping is underdamped, critically damped, or overdamped.

在电路中，基尔霍夫电压定律应用于 RL 电路得到 L(di/dt) + Ri = E(t)，这是一阶线性方程，用积分因子求解。在力学中，胡克定律与牛顿第二定律结合给出阻尼谐振子方程 mx” + cx’ + kx = 0，其解的行为取决于阻尼是欠阻尼、临界阻尼还是过阻尼。

When setting up a model, remember to clearly define variables, state initial or boundary conditions, and verify that the mathematical solution makes physical sense within the context of the problem.

建立模型时，务必清晰定义变量、注明初始条件或边界条件，并检验数学解在问题背景中是否具有物理意义。

12. Exam Strategies and Common Pitfalls | 应试策略与常见错误

Success in ODE questions comes from systematic practice and attention to detail. Always check the type of equation first: is it separable, linear, exact, or reducible? This classification determines the appropriate method, and rushing this step is the most common cause of errors.

解微分方程题目的成功源于系统训练和关注细节。首先判断方程类型：是可分离变量、线性、恰当还是可降阶？这一分类决定了适用方法，而匆忙跳过这一步是出错的最常见原因。

Common pitfalls include forgetting the constant of integration, dropping absolute values when integrating 1/y, failing to multiply by x in undetermined coefficients when there is overlap with the complementary function, and ignoring initial conditions when finding particular solutions. In matrix systems, check that the eigenvectors are correct and remember that complex eigenvalues produce real solutions through real and imaginary parts.

常见错误包括：忘记积分常数；积分 1/y 时去掉绝对值符号；待定系数法中遇到与补函数重叠时未乘以 x；求特解时忽略初始条件。在矩阵系统中，要检验特征向量是否正确，并记住复特征值通过实部和虚部产生实解。

Finally, always test your solution by substituting it back into the original differential equation — a quick check that often catches algebraic slips before you lose marks.

最后，务必把解代回原方程检验——这个快速检查常常能在扣分之前发现代数失误。

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Methods for Solving Ordinary Differential Equations | 常微分方程求解方法