常微分方程简介
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本文内容主要参考自 ODE-The Bright Side of Mathematics
1. 常微分方程的定义
k阶常微分方程 (ODE of order k):
\[ F(t, x, \dot{x}, ..., x^{(k)}) = 0 \]
1.1 autonomous ODE
- autonomous ODE: \(\dot{x} = w(t, x)\)
- non-autonomous ODE: \(\dot{x} = v(x)\)
1.2 homogeneous ODE
- homogeneous ODE: \(\dot{x} = A(t)v(x)\)
- non-homogeneous ODE: \(\dot{x} = A(t)v(x) + B(t)\)
1.3 linear ODE
- linear ODE: \(\dot{x} = A(t)x + B(t)\)