Differential Equations
Core Idea
A differential equation constrains an unknown function through its derivatives.
微分方程描述 quantity 怎样变化，而不直接列出每个时刻的值。求解时，unknown 通常是 function 或 trajectory，不是单个 number。
ODE 使用 ordinary derivatives，通常只有一个 independent variable；PDE 使用 partial derivatives，处理多个 independent variables 或连续分布的 field。
Why It Matters
现实模型常先给 local law：force 决定 acceleration、population size 决定 growth rate、temperature differences 决定 heat flow。微分方程把这些 local rules 连接成 global behavior。
Example
它不是直接给 pendulum angle，而是告诉 angle、angular velocity 与 angular acceleration 必须怎样互相约束。
Connections
- Derivatives supply rates of change.
- Initial Conditions select one member of a solution family.
- Phase Space represents all possible states geometrically.
- Vector Field encodes the change law at every state.
- Euler Method approximates trajectories numerically.
- RC charging turns a loop law into a first-order differential equation.
- RC Circuits
- Electromagnetic damping produces a first-order velocity equation.
- Electromagnetic Damping