有三四周的课没听……基本就是一个速成版速成笔记
Ch1 Introduction
1.1 Differential Equations and Basic Concepts
Def
- linear, nonlinear
- order
- homogeneous, have constant coefficients
Theorem (ODE 的等价形式)
- All ODEs can be equivalently described by a system of 1st-order ODE.
\(\Leftrightarrow\)
\(\Leftrightarrow\)
Tool
- direction field (\(\frac{dy}{dt}=f(t,y)\), draw \((1,f)\) at \((t,y)\))
- integral curve (tangent to \((1,f)\))
1.2 Examples of Solving Tricks
Integrating Factor
\(\Rightarrow\)
- A Necessary Condition:
- For
We need
- If this doesn't work, try multiply an integrating factor:
We need
- For