Separation of Variables

Core Sentence / 核心句

Separation of variables rewrites a separable differential equation so each side contains only one variable before integration.

\frac{d y}{d x} = g (x) h (y) ⟹ \frac{d y}{h ( y )} = g (x) d x .

Separate $x, y$ -> integrate both sides -> use the initial condition -> solve for $y$ -> select the correct branch/domain.

除以 $h (y)$ 时可能丢掉满足 $h (y) = 0$ 的 equilibrium solutions； $ln ∣ \cdot ∣$ 与平方根分支也必须由 initial condition 处理。