一阶微分方程

First-Order Differential Equation

一阶微分方程的一般形式：

\begin{array}{r} y^{'} = f (x, y) \end{array}

\begin{array}{r} g (y) d y = f (x) d x \Rightarrow \int g (y) d y = \int f (x) d x \Rightarrow G (y) = F (x) + C \end{array}

\begin{array}{r} \frac{d y}{d x} = φ (\frac{y}{x}) \end{array}

变量代换： $u = \frac{y}{x}, y = u x$ 求导得到：

\frac{d y}{d x} = u + x \frac{d u}{d x} = φ (u)

分离变量得：

\frac{d u}{φ (u) - u} = \frac{d x}{x}

进一步积分即可得到通解：

\begin{array}{r} \int \frac{d u}{φ (u) - u} = \int \frac{d x}{x} \end{array}

\begin{array}{r} \frac{d y}{d x} = \frac{a x + b y + c}{a_{1} x + b_{1} y + c_{1}} \end{array}

$h, k$ 为待定系数，变量代换： $x = X + h, y = Y + k$ 求导得到： $d x = d X, d y = d Y$

{\begin{cases} a h + b k + c = 0 \\ a_{1} h + b_{1} k + c_{1} = 0 \end{cases}

进一步得到齐次方程：

\begin{array}{r} \frac{d Y}{d X} = \frac{a X + b Y + a h + b k + c}{a_{1} X + b_{1} Y + a_{1} h + b_{1} k + c_{1}} = \frac{a X + b Y}{a_{1} X + b_{1} Y} \end{array}

如果 $\frac{a_{1}}{a} = \frac{b_{1}}{b}$ , 令 $\frac{a_{1}}{a} = \frac{b_{1}}{b} = λ$

变量代换： $v = a x + b y$ 求导得到：

\frac{d v}{d x} = a + b \frac{d y}{d x} \Rightarrow \frac{d y}{d x} = \frac{1}{b} (\frac{d v}{d x} - a)

得到可分离变量的方程：

\begin{array}{r} \frac{d y}{d x} = \frac{a x + b y + c}{λ (a x + b y) + c_{1}} \Rightarrow \frac{1}{b} (\frac{d v}{d x} - a) = \frac{v + c}{λ v + c_{1}} \end{array}