积分与微分方程
积分
不定积分
不定积分的三种求法
∫ g ( ψ ( x ) ) ψ ′ ( x ) d x = ∫ g ( u ) d u ∣ u = ψ ( x ) \int g(\psi(x))\psi'(x) \mathop{}\!\mathrm{d} x = \int g(u) \mathop{}\!\mathrm{d} u \Big|_{u = \psi(x)} ∫ g ( ψ ( x )) ψ ′ ( x ) d x = ∫ g ( u ) d u u = ψ ( x ) 令 x = ψ ( t ) x = \psi(t) x = ψ ( t )
∫ g ( x ) d x = ∫ g ( ψ ( x ) ) d ψ ( x ) = ∫ g ( ψ ( x ) ) ψ ′ ( x ) d x \int g(x) d x = \int g(\psi(x)) d \psi(x) = \int g(\psi(x)) \psi'(x) d x ∫ g ( x ) d x = ∫ g ( ψ ( x )) d ψ ( x ) = ∫ g ( ψ ( x )) ψ ′ ( x ) d x ps: 此法要求 x = ψ ( u ) x = \psi (u) x = ψ ( u ) u = ψ − 1 ( x ) u = \psi^{-1}(x) u = ψ − 1 ( x )
三角函数代换；
根式代换；
倒代换；
复杂函数的直接代换（比如反三角函数和对数函数等）；
**ps:**分布积分经常创造出积分再现或者积分抵消的情形；
∫ u v ′ d x = u v − ∫ v ′ d u \int uv' \mathop{}\!\mathrm{d} x = uv - \int v' \mathop{}\!\mathrm{d} u ∫ u v ′ d x = uv − ∫ v ′ d u 基本积分公式
有理函数
∫ x a d x = x a + 1 a + 1 + C ( a ≠ − 1 ) ; ∫ 1 x d x = ln ∣ x ∣ d x ∫ a x d x = a x ln a d x + C ( a > 0 , a ≠ 1 ) ; ∫ e x d x = e x + C ; \begin{aligned}
&\int x^a \mathop{}\!\mathrm{d} x = \frac{x^{a+1}}{a+1} + C(a \neq -1); &
&\int \frac{1}{x} \mathop{}\!\mathrm{d} x = \ln|x| \mathop{}\!\mathrm{d} x \\
&\int a^x \mathop{}\!\mathrm{d} x = \frac{a^x}{\ln a} \mathop{}\!\mathrm{d} x + C(a > 0, a \neq 1); &
&\int e^x \mathop{}\!\mathrm{d} x = e^x + C; \\
\end{aligned} ∫ x a d x = a + 1 x a + 1 + C ( a = − 1 ) ; ∫ a x d x = ln a a x d x + C ( a > 0 , a = 1 ) ; ∫ x 1 d x = ln ∣ x ∣ d x ∫ e x d x = e x + C ;
三角函数
∫ sin x d x = − cos x + C ; ∫ cos d x = sin x + C ; ∫ tan x d x = − ln ∣ cos x ∣ + C ; ∫ cot x d x = ln sin x + C ; ∫ sec x d x = ln ∣ sec x + tan x ∣ + C ; ∫ csc x d x = ln ∣ csc x − cot x ∣ + C ; ∫ sec 2 x d x = tan x + C ∫ csc 2 x d x = − cot x + ∫ sec x tan x d x = sec x + C ∫ csc x cot x d x = − csc x + C ∫ cos 2 x d x = x 2 − sin 2 x 4 + C ∫ sin 2 x d x = x 2 + sin 2 x 4 + C ∫ tan 2 x d x = tan x − x + C ∫ cot 2 x d x = − cot x − x + C \begin{aligned}
&\int \sin x \mathop{}\!\mathrm{d} x = - \cos x + C; &
&\int \cos \mathop{}\!\mathrm{d} x = \sin x + C; \\
&\int \tan x \mathop{}\!\mathrm{d} x = -\ln |\cos x| + C; &
&\int \cot x \mathop{}\!\mathrm{d} x = \ln \sin x + C; \\
&\int \sec x \mathop{}\!\mathrm{d} x = \ln |\sec x + \tan x| + C; &
&\int \csc x \mathop{}\!\mathrm{d} x = \ln |\csc x - \cot x| + C; \\
&\int \sec^2 x \mathop{}\!\mathrm{d} x = \tan x + C &
&\int \csc^2 x \mathop{}\!\mathrm{d} x = -\cot x + \\
&\int \sec x \tan x \mathop{}\!\mathrm{d} x = \sec x + C&
&\int \csc x \cot x \mathop{}\!\mathrm{d} x = -\csc x + C \\
&\int \cos^2 x \mathop{}\!\mathrm{d} x = \frac{x}{2} - \frac{\sin 2x}{4} + C&
&\int \sin^2 x \mathop{}\!\mathrm{d} x = \frac{x}{2} + \frac{\sin 2x}{4} + C\\
&\int \tan^2 x \mathop{}\!\mathrm{d} x = \tan x - x + C&
&\int \cot^2 x \mathop{}\!\mathrm{d} x = - \cot x - x + C\\
\end{aligned} ∫ sin x d x = − cos x + C ; ∫ tan x d x = − ln ∣ cos x ∣ + C ; ∫ sec x d x = ln ∣ sec x + tan x ∣ + C ; ∫ sec 2 x d x = tan x + C ∫ sec x tan x d x = sec x + C ∫ cos 2 x d x = 2 x − 4 sin 2 x + C ∫ tan 2 x d x = tan x − x + C ∫ cos d x = sin x + C ; ∫ cot x d x = ln sin x + C ; ∫ csc x d x = ln ∣ csc x − cot x ∣ + C ; ∫ csc 2 x d x = − cot x + ∫ csc x cot x d x = − csc x + C ∫ sin 2 x d x = 2 x + 4 sin 2 x + C ∫ cot 2 x d x = − cot x − x + C
sin 2 x , cos 2 x \sin^2 x , \cos^2 x sin 2 x , cos 2 x cos 2 x = 2 cos 2 x − 1 = 1 − 2 sin 2 x \cos 2x = 2\cos^2 x - 1 = 1 - 2\sin^2 x cos 2 x = 2 cos 2 x − 1 = 1 − 2 sin 2 x tan 2 x , cot 2 x \tan^2 x , \cot^2 x tan 2 x , cot 2 x tan 2 = sec 2 x − 1 , cot 2 = csc 2 x − 1 \tan^2 = \sec^2 x - 1, \cot^2 = \csc^2 x - 1 tan 2 = sec 2 x − 1 , cot 2 = csc 2 x − 1
根式
∫ d x 1 − x 2 = arcsin x + C ∫ d x a 2 − x 2 = arcsin x a + C ∫ d x 1 + x 2 = arctan x + C ∫ d x a 2 + x 2 = 1 a arctan x a + C ∫ d x x 2 − a 2 = 1 2 a ln ∣ x − a x + a ∣ + C ∫ d x x 2 ± a 2 = ln ( x + a 2 ± x 2 ) + C ∫ a 2 − x 2 d x = a 2 2 arcsin x a + x 2 a 2 − x 2 + C \begin{aligned}
&\int \frac{\mathop{}\!\mathrm{d} x}{\sqrt{1 - x^2}} = \arcsin x + C&
&\int \frac{\mathop{}\!\mathrm{d} x}{\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} + C\\
&\int \frac{\mathop{}\!\mathrm{d} x}{1 + x^2} = \arctan x + C&
&\int \frac{\mathop{}\!\mathrm{d} x}{a^2 + x^2} = \frac{1}{a} \arctan \frac{x}{a} + C\\
&\int \frac{\mathop{}\!\mathrm{d} x}{x^2 - a^2} = \frac{1}{2a} \ln|\frac{x - a}{x + a}| + C\\
&\int \frac{\mathop{}\!\mathrm{d} x}{\sqrt{x^2 \pm a^2}} = \ln(x + \sqrt{a^2 \pm x^2}) + C\\
&\int \sqrt{a^2 - x^2} \mathop{}\!\mathrm{d} x = \frac{a^2}{2} \arcsin \frac{x}{a} + \frac{x}{2} \sqrt{a^2 - x^2} + C
\end{aligned} ∫ 1 − x 2 d x = arcsin x + C ∫ 1 + x 2 d x = arctan x + C ∫ x 2 − a 2 d x = 2 a 1 ln ∣ x + a x − a ∣ + C ∫ x 2 ± a 2 d x = ln ( x + a 2 ± x 2 ) + C ∫ a 2 − x 2 d x = 2 a 2 arcsin a x + 2 x a 2 − x 2 + C ∫ a 2 − x 2 d x = arcsin a x + C ∫ a 2 + x 2 d x = a 1 arctan a x + C
∫ a x f ( x ) \int^{x}_{a} f(x) ∫ a x f ( x ) f ( x ) f(x) f ( x ) f ′ ( x ) f'(x) f ′ ( x ) f ( x ) f(x) f ( x ) ∫ 0 x f ( t ) d t \int^{x}_{0}f(t) \mathop{}\!\mathrm{d} t ∫ 0 x f ( t ) d t ∫ a x f ( t ) d t \int^{x}_{a}f(t) \mathop{}\!\mathrm{d} t ∫ a x f ( t ) d t f ′ ( x ) f'(x) f ′ ( x ) 周期函数 周期函数(条件：∫ 0 T f ( t ) d t = 0 \int^{T}_{0}f(t) \mathop{}\!\mathrm{d} t = 0 ∫ 0 T f ( t ) d t = 0 周期函数 周期函数 奇函数 偶函数 偶函数 偶函数 偶函数 奇函数 不确定 奇函数
定积分
积分中值定理
∫ b a f ( x ) d x = f ( ξ ) ( b − a ) \int^{a}_{b} f(x) \mathop{}\!\mathrm{d} x = f(\xi) (b - a) ∫ b a f ( x ) d x = f ( ξ ) ( b − a ) 重要公式总结
区间再现公式
∫ b a f ( x ) d x = ∫ b a f ( a + b − x ) d x ∫ b a f ( x ) d x = 1 2 ∫ a b [ f ( x ) + f ( a + b − x ) ] d x ∫ a b f ( x ) d x = ∫ a a + b 2 [ f ( x ) + f ( a + b − x ) ] d x \begin{aligned}
&\int^{a}_{b} f(x) \mathop{}\!\mathrm{d} x = \int^{a}_{b} f(a + b - x) \mathop{}\!\mathrm{d} x\\
&\int^{a}_{b} f(x) \mathop{}\!\mathrm{d} x = \frac{1}{2} \int^{b}_{a} \lbrack f(x) + f(a + b - x) \rbrack \mathop{}\!\mathrm{d} x\\
&\int^{b}_{a} f(x) \mathop{}\!\mathrm{d} x = \int^{\frac{a + b}{2}}_{a} \lbrack f(x) + f(a + b - x) \rbrack \mathop{}\!\mathrm{d} x\\
\end{aligned} ∫ b a f ( x ) d x = ∫ b a f ( a + b − x ) d x ∫ b a f ( x ) d x = 2 1 ∫ a b [ f ( x ) + f ( a + b − x )] d x ∫ a b f ( x ) d x = ∫ a 2 a + b [ f ( x ) + f ( a + b − x )] d x 华氏公式（点火公式）
∫ 0 π 2 sin n x d x = ∫ 0 π 2 cos n x d x = { n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ 2 3 ⋅ 1 , n > 1 and n is odd n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ 2 3 ⋅ 1 , n > 0 and n is even ∫ 0 π sin n x d x = { 2 ⋅ n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 2 3 ⋅ 1 , n > 1 and n is odd 2 ⋅ n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 1 2 ⋅ π 2 , n > 0 and n is even ∫ 0 π cos n x d x = { 0 , n > 0 and n is odd 2 ⋅ n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 1 2 ⋅ π 2 , n > 0 and n is even ∫ 0 2 π sin n x d x = { 0 , n > 0 and n is odd 4 ⋅ n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 1 2 ⋅ π 2 , n > 0 and n is even ∫ 0 2 π cos n x d x = ∫ 0 2 π sin n x d x = { 0 , n > 0 and n is odd 4 ⋅ n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 1 2 ⋅ π 2 , n > 0 and n is even \begin{aligned}
&\int^{\frac{\pi}{2}}_{0} \sin^n x \mathop{}\!\mathrm{d} x = \int^{\frac{\pi}{2}}_{0} \cos^n x \mathop{}\!\mathrm{d} x =
\begin{cases}
\dfrac{n - 1}{n} \cdot \dfrac{n - 3}{n - 2} \cdot \cdots \dfrac{2}{3} \cdot 1, \text{n > 1 and n is odd}\\
\dfrac{n - 1}{n} \cdot \dfrac{n - 3}{n - 2} \cdot \cdots \dfrac{2}{3} \cdot 1, \text{n > 0 and n is even}\\
\end{cases}\\
&\int^{\pi}_{0} \sin^n x \mathop{}\!\mathrm{d} x =
\begin{cases}
2 \cdot \dfrac{n - 1}{n} \cdot \dfrac{n - 3}{n - 2} \cdot \cdots \cdot \dfrac{2}{3} \cdot 1, \text{n > 1 and n is odd}\\
2 \cdot \dfrac{n - 1}{n} \cdot \dfrac{n - 3}{n - 2} \cdot \cdots \cdot \dfrac{1}{2} \cdot \frac{\pi}{2}, \text{n > 0 and n is even}
\end{cases}\\
&\int^{\pi}_{0} \cos^n x \mathop{}\!\mathrm{d} x =
\begin{cases}
0, \text{n > 0 and n is odd}\\
2 \cdot \dfrac{n - 1}{n} \cdot \dfrac{n - 3}{n - 2} \cdot \cdots \cdot \dfrac{1}{2} \cdot \dfrac{\pi}{2}, \text{n > 0 and n is even}
\end{cases}\\
&\int^{2\pi}_{0} \sin^n x \mathop{}\!\mathrm{d} x =
\begin{cases}
0, \text{n > 0 and n is odd}\\
4 \cdot \dfrac{n - 1}{n} \cdot \dfrac{n - 3}{n - 2} \cdot \cdots \cdot \dfrac{1}{2} \cdot \dfrac{\pi}{2}, \text{n > 0 and n is even}
\end{cases}\\
&\int^{2\pi}_{0} \cos^n x \mathop{}\!\mathrm{d} x = \int^{2\pi}_{0} \sin^n x \mathop{}\!\mathrm{d} x =
\begin{cases}
0, \text{n > 0 and n is odd}\\
4 \cdot \dfrac{n - 1}{n} \cdot \dfrac{n - 3}{n - 2} \cdot \cdots \cdot \dfrac{1}{2} \cdot \dfrac{\pi}{2}, \text{n > 0 and n is even}
\end{cases}\\
\end{aligned} ∫ 0 2 π sin n x d x = ∫ 0 2 π cos n x d x = ⎩ ⎨ ⎧ n n − 1 ⋅ n − 2 n − 3 ⋅ ⋯ 3 2 ⋅ 1 , n > 1 and n is odd n n − 1 ⋅ n − 2 n − 3 ⋅ ⋯ 3 2 ⋅ 1 , n > 0 and n is even ∫ 0 π sin n x d x = ⎩ ⎨ ⎧ 2 ⋅ n n − 1 ⋅ n − 2 n − 3 ⋅ ⋯ ⋅ 3 2 ⋅ 1 , n > 1 and n is odd 2 ⋅ n n − 1 ⋅ n − 2 n − 3 ⋅ ⋯ ⋅ 2 1 ⋅ 2 π , n > 0 and n is even ∫ 0 π cos n x d x = ⎩ ⎨ ⎧ 0 , n > 0 and n is odd 2 ⋅ n n − 1 ⋅ n − 2 n − 3 ⋅ ⋯ ⋅ 2 1 ⋅ 2 π , n > 0 and n is even ∫ 0 2 π sin n x d x = ⎩ ⎨ ⎧ 0 , n > 0 and n is odd 4 ⋅ n n − 1 ⋅ n − 2 n − 3 ⋅ ⋯ ⋅ 2 1 ⋅ 2 π , n > 0 and n is even ∫ 0 2 π cos n x d x = ∫ 0 2 π sin n x d x = ⎩ ⎨ ⎧ 0 , n > 0 and n is odd 4 ⋅ n n − 1 ⋅ n − 2 n − 3 ⋅ ⋯ ⋅ 2 1 ⋅ 2 π , n > 0 and n is even 其他含有三角函数的公式
∫ 0 π x f ( sin x ) d x = π 2 ∫ 0 π f ( sin x ) d x ∫ 0 π x f ( sin x ) d x = π ∫ 0 π 2 f ( sin x ) d x ∫ 0 π 2 f ( sin x ) d x = ∫ 0 π 2 f ( cos x ) d x ∫ 0 π 2 f ( sin x , cos x ) d x = ∫ 0 π 2 f ( cos x , sin x ) d x \begin{aligned}
&\int^{\pi}_{0} x f(\sin x) \mathop{}\!\mathrm{d} x = \dfrac{\pi}{2} \int^{\pi}_{0} f(\sin x ) \mathop{}\!\mathrm{d} x\\
&\int^{\pi}_{0} x f(\sin x) \mathop{}\!\mathrm{d} x = \pi \int^{\frac{\pi}{2}}_{0} f(\sin x) \mathop{}\!\mathrm{d} x\\
&\int^{\frac{\pi}{2}}_{0} f(\sin x) \mathop{}\!\mathrm{d} x = \int^{\frac{\pi}{2}}_{0} f(\cos x) \mathop{}\!\mathrm{d} x\\
&\int^{\frac{\pi}{2}}_{0} f(\sin x, \cos x) \mathop{}\!\mathrm{d} x = \int^{\frac{\pi}{2}}_{0} f(\cos x, \sin x) \mathop{}\!\mathrm{d} x
\end{aligned} ∫ 0 π x f ( sin x ) d x = 2 π ∫ 0 π f ( sin x ) d x ∫ 0 π x f ( sin x ) d x = π ∫ 0 2 π f ( sin x ) d x ∫ 0 2 π f ( sin x ) d x = ∫ 0 2 π f ( cos x ) d x ∫ 0 2 π f ( sin x , cos x ) d x = ∫ 0 2 π f ( cos x , sin x ) d x 区间简化公式
∫ a b f ( x ) d x = ∫ − π 2 π 2 f ( a + b 2 + b − a 2 sin t ) ⋅ b − a 2 cos t d t ∫ a b f ( x ) d x = ∫ 0 1 f [ a + ( b − a ) t ] d t ∫ − a a f ( x ) d x = ∫ 0 a [ f ( x ) + f ( − x ) ] d x \begin{aligned}
&\int^{b}_{a} f(x) \mathop{}\!\mathrm{d} x = \int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} f(\dfrac{a + b}{2} + \frac{b - a}{2} \sin t) \cdot \frac{b - a}{2} \cos t \mathop{}\!\mathrm{d} t\\
&\int^{b}_{a} f(x) \mathop{}\!\mathrm{d} x = \int^{1}_{0} f\lbrack a + (b - a)t \rbrack \mathop{}\!\mathrm{d} t\\
&\int^{a}_{-a} f(x) \mathop{}\!\mathrm{d} x = \int^{a}_{0} \lbrack f(x) + f(-x) \rbrack \mathop{}\!\mathrm{d} x\\
\end{aligned} ∫ a b f ( x ) d x = ∫ − 2 π 2 π f ( 2 a + b + 2 b − a sin t ) ⋅ 2 b − a cos t d t ∫ a b f ( x ) d x = ∫ 0 1 f [ a + ( b − a ) t ] d t ∫ − a a f ( x ) d x = ∫ 0 a [ f ( x ) + f ( − x )] d x 变限积分
直接求导
∫ 0 f ( x ) g ( t ) d t ⇒ g ( f ( x ) ) f ′ ( x ) \int^{f(x)}_{0} g(t) \mathop{}\!\mathrm{d} t \Rightarrow g(f(x))f'(x) ∫ 0 f ( x ) g ( t ) d t ⇒ g ( f ( x )) f ′ ( x )
换元求导
∫ x x + f ( x ) g ( t − x ) d t ⇒ u = t − x = ∫ 0 f ( x ) g ( u ) d u \int^{x + f(x)}_{x} g(t - x) \mathop{}\!\mathrm{d} t \xRightarrow{u = t - x} = \int^{f(x)}_{0} g(u) \mathop{}\!\mathrm{d} u ∫ x x + f ( x ) g ( t − x ) d t u = t − x = ∫ 0 f ( x ) g ( u ) d u
拆分求导，一般是对称区间奇函数或者绝对值等等分段函数
换序积分，画出积分范围，再换序
注意常考 ：
反函数的导函数关系
原函数与积分的关系
积分中值定理
拉格朗日中值定理
含 e x e^x e x
反常积分与 Γ \varGamma Γ
微分方程
可分离变量的微分方程
d y d x = P ( x ) y → d y y = P ( x ) d x → ln y = ∫ P ( x ) d x → y = C e ∫ P ( x ) d x \frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} = P(x) y \rightarrow
\frac{\mathop{}\!\mathrm{d} y}{y} = P(x) \mathop{}\!\mathrm{d} x \to
\ln y = \int P(x) \mathop{}\!\mathrm{d} x \to
y = C e ^{\int P(x) \mathop{}\!\mathrm{d} x} d x d y = P ( x ) y → y d y = P ( x ) d x → ln y = ∫ P ( x ) d x → y = C e ∫ P ( x ) d x 齐次微分方程
d y d x = ϕ ( y x ) ⇒ u = y x d y d x = x d u d x + u = ϕ ( u ) ⇒ d u u − ϕ ( u ) = d x x ⇒ x = C e ∫ d u u − ϕ ( u ) , u = y x \begin{aligned}
& \frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} = \phi(\frac{y}{x}) \xRightarrow{u = \frac{y}{x}}
\frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} = x \frac{\mathop{}\!\mathrm{d} u}{\mathop{}\!\mathrm{d} x} + u = \phi(u) \\
& \Rightarrow \frac{\mathop{}\!\mathrm{d} u}{u - \phi (u)} = \frac{\mathop{}\!\mathrm{d} x}{x} \\
& \Rightarrow x = C e ^{\int \frac{\mathop{}\!\mathrm{d} u}{u - \phi (u)}}, u = \frac{y}{x}
\end{aligned} d x d y = ϕ ( x y ) u = x y d x d y = x d x d u + u = ϕ ( u ) ⇒ u − ϕ ( u ) d u = x d x ⇒ x = C e ∫ u − ϕ ( u ) d u , u = x y 可化为齐次的微分方程
d y d x = a 1 x + b 1 y + c 1 a 2 x = b 2 y + c 2 ⇒ x = X + k 1 , y = Y + k 2 d y d x = d Y d X = a 1 X + b 1 Y a 2 X + b 2 Y ⇒ u = Y X d Y d X = b 1 u + a 1 b 2 u + a 2 \frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} = \frac{a_1 x + b_1 y + c_1}{a_2 x = b_2 y + c_2} \xRightarrow{x = X + k_1, y = Y + k_2}
\frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} = \frac{\mathop{}\!\mathrm{d} Y}{\mathop{}\!\mathrm{d} X} = \frac{a_1 X + b_1 Y}{a_2 X + b_2 Y} \xRightarrow{u = \frac{Y}{X}}
\frac{\mathop{}\!\mathrm{d} Y}{\mathop{}\!\mathrm{d} X} = \frac{b_1 u + a_1}{b_2 u + a_2} d x d y = a 2 x = b 2 y + c 2 a 1 x + b 1 y + c 1 x = X + k 1 , y = Y + k 2 d x d y = d X d Y = a 2 X + b 2 Y a 1 X + b 1 Y u = X Y d X d Y = b 2 u + a 2 b 1 u + a 1 上述 k 1 , k 2 k_1, k_2 k 1 , k 2 c 1 , c 2 c_1, c_2 c 1 , c 2
一阶线性微分方程
有以下一阶线性微分方程：
d y d x + P ( x ) y = Q ( x ) \frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} + P(x)y = Q(x) d x d y + P ( x ) y = Q ( x ) 令 Q ( x ) = 0 Q(x) = 0 Q ( x ) = 0 非齐次微分方程对应的齐次方程 ：
d y d x = − P ( x ) y \frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} = - P(x) y d x d y = − P ( x ) y 齐次方程的通解：
d y d x + P ( x ) y = 0 ⇒ d y y = − P ( x ) d x ⇒ y = C e − ∫ P ( x ) d x \frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} + P(x)y = 0 \Rightarrow
\frac{\mathop{}\!\mathrm{d} y}{y} = -P(x) \mathop{}\!\mathrm{d} x \Rightarrow y = Ce ^{- \int P(x) \mathop{}\!\mathrm{d} x} d x d y + P ( x ) y = 0 ⇒ y d y = − P ( x ) d x ⇒ y = C e − ∫ P ( x ) d x 常数变易法解出非齐次微分方程 的通解：
令 y = u e − ∫ P ( x ) d x y = u e ^{- \int P(x) \mathop{}\!\mathrm{d} x} y = u e − ∫ P ( x ) d x
d y d x = − P ( x ) u e − ∫ P ( x ) d x + d u d x e − ∫ P ( x ) d x = Q ( x ) − P ( x ) y = Q ( x ) − P ( x ) u e − ∫ P ( x ) d x ⟹ d u d x = Q ( x ) e ∫ P ( x ) d x ⟹ u = ∫ Q ( x ) e ∫ P ( x ) d x d x + C ⟹ y = ( ∫ Q ( x ) e ∫ P ( x ) d x d x + C ) e − ∫ P ( x ) d x \begin{aligned}
\frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} & = -P(x) u e ^{- \int P(x) \mathop{}\!\mathrm{d} x} + \frac{\mathop{}\!\mathrm{d} u}{\mathop{}\!\mathrm{d} x} e ^{-\int P(x) \mathop{}\!\mathrm{d} x} \\
& = Q(x) - P(x) y \\
& = Q(x) - P(x) u e ^{-\int P(x) \mathop{}\!\mathrm{d} x} \\ \Longrightarrow
& \frac{\mathop{}\!\mathrm{d} u}{\mathop{}\!\mathrm{d} x} = Q(x) e ^{\int P(x) \mathop{}\!\mathrm{d} x} \\ \Longrightarrow
& u = \int Q(x) e ^{\int P(x) \mathop{}\!\mathrm{d} x} \mathop{}\!\mathrm{d} x + C \\ \Longrightarrow
& y = \big( \int Q(x) e ^{\int P(x) \mathop{}\!\mathrm{d} x} \mathop{}\!\mathrm{d} x + C \big) e ^{- \int P(x) \mathop{}\!\mathrm{d} x}
\end{aligned} d x d y ⟹ ⟹ ⟹ = − P ( x ) u e − ∫ P ( x ) d x + d x d u e − ∫ P ( x ) d x = Q ( x ) − P ( x ) y = Q ( x ) − P ( x ) u e − ∫ P ( x ) d x d x d u = Q ( x ) e ∫ P ( x ) d x u = ∫ Q ( x ) e ∫ P ( x ) d x d x + C y = ( ∫ Q ( x ) e ∫ P ( x ) d x d x + C ) e − ∫ P ( x ) d x 伯努利方程
d y d x + P ( x ) y = Q ( x ) y n ⇒ y − n d y d x + P ( x ) y 1 − n = Q ( x ) ⇒ d y 1 − n d x + ( 1 − n ) P ( x ) y 1 − n = ( 1 − n ) Q ( x ) \begin{aligned}
& \frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} + P(x)y = Q(x)y ^{n} \\ \Rightarrow
& y^{-n} \frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} + P(x) y^{1-n} = Q(x) \\ \Rightarrow
& \frac{\mathop{}\!\mathrm{d} y^{1-n}}{\mathop{}\!\mathrm{d} x} + (1-n)P(x)y^{1-n} = (1-n)Q(x)
\end{aligned} ⇒ ⇒ d x d y + P ( x ) y = Q ( x ) y n y − n d x d y + P ( x ) y 1 − n = Q ( x ) d x d y 1 − n + ( 1 − n ) P ( x ) y 1 − n = ( 1 − n ) Q ( x ) 最后化成了关于 y 1 − n y^{1-n} y 1 − n x x x y 1 − n y^{1-n} y 1 − n y y y
可降阶的高阶微分方程
y ( n ) = f ( x ) y^{(n)} = f(x) y ( n ) = f ( x ) 无限求积分直到解出来
y = ∫ ( ∫ … ( ∫ ( ∫ y ( n ) d x + C 1 ) d x + C 2 ) + … ) + C n − 1 y = \int \Big( \int \dots \big( \int (\int y ^{(n)} \mathop{}\!\mathrm{d} x + C_1 ) \mathop{}\!\mathrm{d} x + C_2 \big) + \dots \Big) + C_{n-1} y = ∫ ( ∫ … ( ∫ ( ∫ y ( n ) d x + C 1 ) d x + C 2 ) + … ) + C n − 1 y ′ ′ = f ( x , y ′ ) y'' = f(x, y') y ′′ = f ( x , y ′ ) 该类型的微分方程右端不显含未知函数 y y y y ′ = p y' = p y ′ = p
p ′ = f ( x , p ) p' = f(x, p) p ′ = f ( x , p ) 解该一阶微分方程得到：
p = φ ( x , C 1 ) ⇒ d y d x = φ ( x , C 1 ) d x ⇒ y = ∫ φ ( x , C 1 ) d x \begin{aligned}
p &= \varphi(x, C_1) \\ \Rightarrow
\frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} &= \varphi(x, C_1) \mathop{}\!\mathrm{d} x \\ \Rightarrow
y &= \int \varphi(x, C_1) \mathop{}\!\mathrm{d} x
\end{aligned} p ⇒ d x d y ⇒ y = φ ( x , C 1 ) = φ ( x , C 1 ) d x = ∫ φ ( x , C 1 ) d x y ′ ′ = f ( y , y ′ ) y'' = f(y, y') y ′′ = f ( y , y ′ ) 该方程不显含自变量 x x x p = y ′ p = y' p = y ′ y y y p p p
y ′ ′ = d y ′ d y ⋅ d y d x = p ⋅ d p d x = f ( y , p ) y'' = \frac{\mathop{}\!\mathrm{d} y'}{\mathop{}\!\mathrm{d} y} \cdot \frac{\mathop{}\!\mathrm{d} y}{\mathop{}\!\mathrm{d} x} = p \cdot \frac{\mathop{}\!\mathrm{d} p}{\mathop{}\!\mathrm{d} x} = f(y, p) y ′′ = d y d y ′ ⋅ d x d y = p ⋅ d x d p = f ( y , p ) 这是一个关于 p p p y y y
y ′ = p = φ ( y , C 1 ) ⇒ ∫ d y φ ( y , C 1 ) = x + C 2 \begin{aligned}
y' = p &= \varphi(y, C_1) \\ \Rightarrow
\int \frac{\mathop{}\!\mathrm{d} y}{\varphi (y, C_1)} &= x + C_2
\end{aligned} y ′ = p ⇒ ∫ φ ( y , C 1 ) d y = φ ( y , C 1 ) = x + C 2 高阶微分方程
线性微分方程解的结构
考虑一个二阶线性微分方程：
y ′ ′ + P ( x ) y ′ + Q ( x ) y = 0 (1) \tag{1} y'' + P(x)y' + Q(x)y = 0 y ′′ + P ( x ) y ′ + Q ( x ) y = 0 ( 1 ) 定理 1 如果函数 y 1 ( x ) y_1(x) y 1 ( x ) y 2 ( x ) y_2(x) y 2 ( x ) ( 1 ) (1) ( 1 )
y = C 1 y 1 ( x ) + C 2 y 2 ( x ) y = C_1 y_1(x) + C_2 y_2(x) y = C 1 y 1 ( x ) + C 2 y 2 ( x ) 也是方程 ( 1 ) (1) ( 1 ) C 1 , C 2 C_1, C_2 C 1 , C 2
线性相关的概念：
设函数 y 1 ( x ) , y 2 ( x ) , … , y n ( x ) y_1(x), y_2(x), \dots, y_n(x) y 1 ( x ) , y 2 ( x ) , … , y n ( x ) I I I n n n n n n k 1 , k 2 , k 3 , … , k n k_1, k_2, k_3, \dots, k_n k 1 , k 2 , k 3 , … , k n k 1 y 1 ( x ) + k 2 y 2 ( x ) + ⋯ + k n y n ( x ) ≡ 0 k_1 y_1(x) + k_2 y_2(x) + \dots + k_n y_n(x) \equiv 0 k 1 y 1 ( x ) + k 2 y 2 ( x ) + ⋯ + k n y n ( x ) ≡ 0 n n n y 1 ( x ) , y 2 ( x ) , y 3 ( x ) , … , y n ( x ) y_1(x), y_2(x), y_3(x), \dots, y_n(x) y 1 ( x ) , y 2 ( x ) , y 3 ( x ) , … , y n ( x ) 线性相关 ，否则称为线性无关 .
定理 2 如果 y 1 ( x ) y_1(x) y 1 ( x ) y 2 ( x ) y_2(x) y 2 ( x ) ( 1 ) (1) ( 1 )
y = C 1 y 1 ( x ) + C 2 y 2 ( x ) ( C 1 , C 2 are constants ) y = C_1 y_1(x) + C_2 y_2(x) \quad (C_1, C_2 \; \text{are constants}) y = C 1 y 1 ( x ) + C 2 y 2 ( x ) ( C 1 , C 2 are constants ) 就是方程 ( 1 ) (1) ( 1 )
… 未完待续
常系数齐次线性微分方程
常系数非齐次线性微分方程
欧拉方程