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常微分方程整理

一阶线性微分方程

变量可分离方程

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形式: $$ \frac{\d y}{\d x} = f(x)g(y) $$

解法: 分离变量再积分

$$ \int \frac{1}{g(y)} \d y = \int f(x)\d x $$

齐次方程

形式 1: $$ \frac{\d y}{\d x} = g(\frac{y}{x}) $$

解法:

$$ \text{令} u = \frac{y}{x} \text{则,} $$

$$ u = \frac{y}{x} \Rightarrow y = u x \Rightarrow \frac{\d y}{\d x} = u + x \frac{\d y}{\d x} $$

代回原式,即可化为变量可分离方程

$$ \begin{align} u + x \frac{\d u}{\d x} & = g(u) \\ \frac{\d u}{\d x} &= \frac{g(u)-u}{x} \end{align} $$

解出后,将 $ u = \frac{y}{x} $ 代回即可

形式 2: $$ \frac{\d y}{\d x} = \frac{ a_{1} x + b_{1} y + c_{1} }{ a_{2} x + b_{2} y + c_{2} } $$

解法:

情况 1: 若 $$ \frac{ a_{1} }{ a_{2} } = \frac{ b_{1} }{ b_{2} } = \frac{ c_{1} }{ c_{2} } = k $$ 则 $$ \frac{ \d y }{ \d x } = k \Rightarrow y = kx + c $$

情况 2: 若 $$ \frac{ a_{1} }{ a_{2} } = \frac{ b_{1} }{ b_{2} } \neq \frac{ c_{1} }{ c_{2} } $$

线性微分方程

恰当方程

隐式方程

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