📚 Solving Differential Equations: Basic Methods | 求解微分方程的基本方法
A differential equation is a mathematical equation that relates a function with its derivatives. In A-level mathematics, mastering the basic methods for solving differential equations is essential for modelling real-world phenomena, from population growth to radioactive decay.
微分方程是将函数与其导数联系起来的数学方程。在A-level数学中，掌握求解微分方程的基本方法，是对人口增长、放射性衰变等现实世界现象建模的关键。
1. What Is a Differential Equation? | 什么是微分方程？
A differential equation contains one or more derivatives of an unknown function. The order of a differential equation is determined by the highest derivative present. For example, dy/dx = 3x is first-order, while d²y/dx² + y = 0 is second-order.
微分方程包含未知函数的一个或多个导数。微分方程的阶由最高阶导数决定。例如，dy/dx = 3x 是一阶方程，而 d²y/dx² + y = 0 是二阶方程。
The solution of a differential equation is a function, not a number. A general solution contains arbitrary constants, whereas a particular solution is obtained when extra conditions are specified.
微分方程的解是函数而不是数。通解包含任意常数，而当给定额外条件时得到特解。
2. Separable Variables | 变量分离法
A first-order differential equation is separable if it can be written in the form dy/dx = f(x)g(y). The method involves rearranging the equation so that all terms involving y are on one side and all terms involving x are on the other side, then integrating both sides.
一阶微分方程若能写成 dy/dx = f(x)g(y) 的形式，则可使用变量分离法。该方法通过重排方程，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。
For example, to solve dy/dx = x/y, we rewrite as y dy = x dx. Integrating gives ½y² = ½x² + C, which simplifies to y² – x² = C.
例如，求解 dy/dx = x/y，可改写为 y dy = x dx。积分得 ½y² = ½x² + C，即 y² – x² = C。
∫ y dy = ∫ x dx → ½y² = ½x² + C
Remember to include the constant of integration at the end of the side with the independent variable. Never skip this step, as it determines the family of solutions.
记得在自变量所在的那一侧加上积分常数。千万不要省略这一步，因为它决定了解的族。
3. First-Order Linear Equations and the Integrating Factor | 一阶线性方程与积分因子
A first-order linear differential equation has the form dy/dx + P(x)y = Q(x). To solve it, we multiply both sides by an integrating factor I = e^(∫P dx). This transforms the left-hand side into the derivative of I·y.
一阶线性微分方程具有形式 dy/dx + P(x)y = Q(x)。求解时，我们两边乘上积分因子 I = e^(∫P dx)，这会将左边化为 I·y 的导数。
The standard procedure is: first find the integrating factor, then multiply the equation, integrate both sides, and finally rearrange to obtain y as a function of x.
标准步骤是：先求积分因子，然后乘到方程两边，两边积分，最后整理得到 y 关于 x 的表达式。
I(x) = e^(∫P(x) dx), then d/dx (I·y) = I·Q(x)
For example, for dy/dx + 2y = e⁻ˣ, P(x) = 2, so I = e^(2x). Multiplying gives e^(2x)dy/dx + 2e^(2x)y = eˣ, hence d/dx(e^(2x)y) = eˣ.
例如，对于 dy/dx + 2y = e⁻ˣ，P(x) = 2，所以 I = e^(2x)。两边相乘得 e^(2x)dy/dx + 2e^(2x)y = eˣ，因此 d/dx(e^(2x)y) = eˣ。
4. Homogeneous Differential Equations | 齐次微分方程
A differential equation is homogeneous if it can be written in the form dy/dx = F(y/x). These equations are solved by substituting y = vx, where v is a new dependent variable. Then dy/dx = v + x dv/dx, which often makes the equation separable.
可写成 dy/dx = F(y/x) 形式的微分方程称为齐次方程。求解时令 y = vx，其中 v 是新的因变量。此时 dy/dx = v + x dv/dx，通常可使方程变为可分离变量形式。
After substitution, separate the variables in v and x, integrate, and then replace v by y/x to return to the original variables.
代入后，分离 v 与 x 的变量，积分，再用 v = y/x 代回原变量。
Let y = vx, dy/dx = v + x dv/dx
This method is particularly useful for equations where the right-hand side is a ratio of homogeneous polynomials of the same degree.
当方程右侧是同次齐次多项式之比时，这种方法特别有效。
5. Exact Differential Equations | 恰当微分方程
An equation of the form M(x,y) dx + N(x,y) dy = 0 is exact if ∂M/∂y = ∂N/∂x. If this condition holds, there exists a function F(x,y) such that dF = M dx + N dy, and the solution is F(x,y) = C.
形如 M(x,y) dx + N(x,y) dy = 0 的方程，若满足 ∂M/∂y = ∂N/∂x，则称为恰当方程。此时存在函数 F(x,y)，使 dF = M dx + N dy，其解为 F(x,y) = C。
To find F, integrate M with respect to x (or N with respect to y), then determine the arbitrary function of the other variable using the remaining equation. Finally, write the solution in implicit form.
求 F 时，先对 x 积分 M（或对 y 积分 N），再利用另一个方程确定关于另一个变量的任意函数，最后写出隐式解。
If the equation is not exact, it may sometimes be made exact by multiplying by an integrating factor. However, finding such a factor is often more advanced than the basic A-level syllabus.
若方程不是恰当方程，有时可通过乘以积分因子使其变为恰当方程。不过，求积分因子通常超出A-level基础大纲的范围。
6. Second-Order Linear Equations with Constant Coefficients | 常系数二阶线性方程
A second-order linear differential equation with constant coefficients has the form a d²y/dx² + b dy/dx + c y = f(x). If f(x) = 0, the equation is homogeneous; otherwise it is non-homogeneous.
常系数二阶线性微分方程具有形式 a d²y/dx² + b dy/dx + c y = f(x)。若 f(x) = 0，方程为齐次的；否则为非齐次。
For the homogeneous case, we assume a solution of the form y = e^(kx). Substituting into the equation gives the auxiliary equation a k² + b k + c = 0. The roots of this quadratic determine the general solution.
对于齐次情形，我们假设解具有形式 y = e^(kx)。代入方程得到辅助方程 a k² + b k + c = 0。该二次方程的根决定了通解的形式。
a k² + b k + c = 0
The three possible cases for the roots are: real and distinct, real and equal, and complex conjugates. Each case has its own standard form of solution.
根有三种可能：相异实根、相等的实根、共轭复根。每种情形都有各自的标准解形式。
7. Real and Distinct Roots | 相异实根
If the auxiliary equation has two real and distinct roots k₁ and k₂, then the general solution of the homogeneous equation is y = A e^(k₁x) + B e^(k₂x), where A and B are arbitrary constants.
若辅助方程有两个相异实根 k₁ 和 k₂，则齐次方程的通解为 y = A e^(k₁x) + B e^(k₂x)，其中 A、B 为任意常数。
For example, if the auxiliary equation is k² – 3k + 2 = 0, the roots are k = 1 and k = 2, so the solution is y = A eˣ + B e^(2x).
例如，若辅助方程为 k² – 3k + 2 = 0，根为 k = 1 和 k = 2，则解为 y = A eˣ + B e^(2x)。
This solution represents a linear combination of two exponential functions, each corresponding to a root of the auxiliary equation.
该解是两个指数函数的线性组合，每个指数函数对应辅助方程的一个根。
8. Repeated Roots and Complex Roots | 重根与复根
If the auxiliary equation has a repeated root k, the general solution is y = (A + Bx)e^(kx). The extra factor x arises because one exponential alone cannot account for two linearly independent solutions.
若辅助方程有重根 k，通解为 y = (A + Bx)e^(kx)。额外的因子 x 出现是因为单个指数不足以提供两个线性无关的解。
If the roots are complex and of the form k = p ± qi, the general solution is y = e^(px)(A cos(qx) + B sin(qx)). This can also be written using the exponential form of complex numbers.
若根为复数且形如 k = p ± qi，通解为 y = e^(px)(A cos(qx) + B sin(qx))。也可利用复数的指数形式表达。
Repeated: y = (A + Bx)e^(kx) | Complex: y = e^(px)(A cos qx + B sin qx)
For instance, if the auxiliary equation is k² + 4k + 5 = 0, the roots are -2 ± i, so the solution is y = e^(-2x)(A cos x + B sin x).
例如，若辅助方程为 k² + 4k + 5 = 0，根为 -2 ± i，则解为 y = e^(-2x)(A cos x + B sin x)。
9. Particular Integral and Complementary Function | 特解与补函数
For a non-homogeneous equation a d²y/dx² + b dy/dx + c y = f(x), the general solution is the sum of the complementary function (CF) and a particular integral (PI). The complementary function is the general solution of the corresponding homogeneous equation.
对非齐次方程 a d²y/dx² + b dy/dx + c y = f(x)，通解等于补函数（CF）与特解（PI）之和。补函数是对应齐次方程的通解。
To find the particular integral, we choose a trial function based on the form of f(x). Common trials are: constant for a constant f(x), A e^(λx) for f(x) = e^(λx), A sin(mx) + B cos(mx) for sinusoidal f(x), and a polynomial of the same degree for polynomial f(x).
求特解时，我们根据 f(x) 的形式选择试探函数。常见试探有：f(x) 为常数时取常数；f(x) = e^(λx) 时取 A e^(λx)；f(x) 为正弦/余弦时取 A sin(mx) + B cos(mx)；f(x) 为多项式时取同次多项式。
If the trial function also satisfies the homogeneous equation, we multiply it by x (or x² for repeated roots) until it is no longer a solution of the homogeneous part. This is called the modification rule.
若试探函数本身就满足齐次方程，则需乘以 x（重根时乘 x²），直到它不再是齐次部分的解。这称为修正规则。
10. Initial Value Problems and Boundary Conditions | 初值问题与边界条件
An initial value problem supplies conditions at a single point, such as y(0) = 1 and dy/dx(0) = 2. These conditions allow us to determine the arbitrary constants in the general solution.
初值问题在单个点给出条件，例如 y(0) = 1 和 dy/dx(0) = 2。这些条件使我们能够确定通解中的任意常数。
Boundary conditions, in contrast, specify values at two different points, for example y(0) = 0 and y(1) = 5. These are common in problems involving beams, heat flow, and other physical systems.
边界条件则在不同点给出取值，例如 y(0) = 0 和 y(1) = 5。这在涉及梁、热流及其他物理系统的问题中很常见。
When applying the conditions, first differentiate the general solution if the derivative is given. Then solve the resulting equations simultaneously to find the constants.
应用条件时，若给出的是导数，需先对通解求导。然后联立所得方程求解常数。
11. Applications in Physics and Engineering | 物理与工程中的应用
Differential equations model many natural processes. For example, Newton’s law of cooling states dT/dt = -k(T – Tₐ), which is separable and gives T = Tₐ + C e^(-kt).
微分方程可模拟许多自然过程。例如，牛顿冷却定律 dT/dt = -k(T – Tₐ) 是可分离变量的，其解为 T = Tₐ + C e^(-kt)。
Simple harmonic motion described by d²x/dt² + ω²x = 0 has the solution x = A cos(ωt) + B sin(ωt). Such equations arise in pendulums, springs, and electrical circuits.
简谐运动满足 d²x/dt² + ω²x = 0，其解为 x = A cos(ωt) + B sin(ωt)。此类方程出现在单摆、弹簧和电路中。
In population dynamics, the logistic equation dP/dt = rP(1 – P/K) is a more advanced separable equation that accounts for limited resources.
在种群动力学中，逻辑斯蒂方程 dP/dt = rP(1 – P/K) 是一个更高级的可分离方程，它考虑了有限资源。
Being able to choose the correct method — separation, integrating factor, or auxiliary equation — is the key skill in modelling applications.
能够选择正确的方法——分离变量、积分因子或辅助方程——是应用建模中的关键技能。
12. Summary of Methods | 方法总结
The table below summarises the methods covered:
下表总结了所介绍的方法：
| Type | Form | Method |
| First-order separable | dy/dx = f(x)g(y) | Separate and integrate |
| First-order linear | dy/dx + P(x)y = Q(x) | Integrating factor e^(∫P dx) |
| Homogeneous | dy/dx = F(y/x) | Substitute y = vx |
| Exact | M dx + N dy = 0 | Check ∂M/∂y = ∂N/∂x |
| Second-order constant coefficients | a y” + b y’ + c y = f(x) | Auxiliary equation + particular integral |
Start by identifying the order and type of the equation. Then apply the appropriate standard procedure. With regular practice, you will be able to recognise the correct method quickly under exam conditions.
首先判断方程的阶和类型，然后应用相应的标准步骤。通过规律练习，在考试中你将能快速识别正确的方法。
Remember that every solution step should be justified. The constant of integration, the domain of the solution, and any assumptions about the variables must always be checked.
记住，每一步求解都应有理有据。积分常数、解的定义域，以及对变量的任何假设都必须始终检查。
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