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小程序网_站长之家seo概况查询_关键词seo优化排名_百度关键词搜索怎么弄

2024/12/23 9:06:34 来源:https://blog.csdn.net/annesede/article/details/142529284  浏览:    关键词:小程序网_站长之家seo概况查询_关键词seo优化排名_百度关键词搜索怎么弄
小程序网_站长之家seo概况查询_关键词seo优化排名_百度关键词搜索怎么弄

P.S. 本来不想分篇的,但居然字数超上限了。

2 π x < sin ⁡ x < x ( 0 < x < π 2 ) . \frac{2}{\pi}x<\sin x<x\ (0<x<\frac{\pi}{2}). π2x<sinx<x (0<x<2π).

x < tan ⁡ x < 4 π x ( 0 < x < π 4 ) . x<\tan{x}<\frac{4}{\pi}x\ (0<x<\frac{\pi}{4}). x<tanx<π4x (0<x<4π).

1 x + 1 < ln ⁡ ( 1 + 1 x ) < 1 x ( x > 0 ) . \frac{1}{x+1}<\ln(1+\frac{1}{x})<\frac{1}{x}\ (x>0). x+11<ln(1+x1)<x1 (x>0).

x 1 + x < ln ⁡ ( 1 + x ) < x ( x > 0 ) . \frac{x}{1+x}<\ln(1+x)<x\ (x>0). 1+xx<ln(1+x)<x (x>0).

arctan ⁡ x ≤ x ≤ arcsin ⁡ x ( 0 ≤ x ≤ 1 ) . \arctan{x}\leq x\leq\arcsin{x} (0\leq x\leq 1). arctanxxarcsinx(0x1).

sin ⁡ ( arccos ⁡ x ) = 1 − x 2 ; tan ⁡ ( arcsin ⁡ x ) = x 1 − x 2 ; sin ⁡ ( arctan ⁡ x ) = x 1 + x 2 . \sin(\arccos{x})=\sqrt{1-x^2};\ \tan(\arcsin{x})=\frac{x}{\sqrt{1-x^2}};\ \sin(\arctan{x})=\frac{x}{\sqrt{1+x^2}}. sin(arccosx)=1x2 ; tan(arcsinx)=1x2 x; sin(arctanx)=1+x2 x.

lim ⁡ u → 1 , v → ∞ u v = exp ⁡ lim ⁡ u → 1 , v → ∞ ( u − 1 ) v . \lim_{u\to 1, v\to\infty}u^v=\exp\lim_{u\to 1, v\to\infty}(u-1)v. u1,vlimuv=expu1,vlim(u1)v.

f ( x ) − f ( x 0 ) = ∑ i = 1 n f ( i ) ( x 0 ) i ! ( x − x 0 ) i + f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( x − x 0 ) n + 1 , ξ ∈ ( x , x 0 ) ∨ ( x 0 , x ) . f(x)-f(x_0)=\sum_{i=1}^n\frac{f^{(i)}(x_0)}{i!}(x-x_0)^i+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1},\ \xi\in(x,x_0)\vee(x_0,x). f(x)f(x0)=i=1ni!f(i)(x0)(xx0)i+(n+1)!f(n+1)(ξ)(xx0)n+1, ξ(x,x0)(x0,x).

sin ⁡ x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + . . . \sin{x}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+... sinx=x3!x3+5!x57!x7+...

cos ⁡ x = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + . . . \cos{x}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+... cosx=12!x2+4!x46!x6+...

tan ⁡ x = x + 1 3 x 3 + . . . \tan x=x+\frac{1}{3}x^3+... tanx=x+31x3+...

arcsin ⁡ x = x + 1 6 x 3 + . . . \arcsin x=x+\frac{1}{6}x^3+... arcsinx=x+61x3+...

arctan ⁡ x = x − 1 3 x 3 + . . . \arctan x=x-\frac{1}{3}x^3+... arctanx=x31x3+...

ln ⁡ ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 − 1 4 x 4 + . . . \ln(1+x)=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4+... ln(1+x)=x21x2+31x341x4+...

( 1 + x ) 1 x = e − e 2 x + . . . (1+x)^\frac{1}{x}=e-\frac{e}{2}x+... (1+x)x1=e2ex+...

1 1 + x n = 1 − x n + x 2 n − x 3 n + . . . \frac{1}{1+x^n}=1-x^n+x^{2n}-x^{3n}+... 1+xn1=1xn+x2nx3n+...

( 1 + x ) a = 1 + a x + a ( a − 1 ) 2 ! x 2 + a ( a − 1 ) ( a − 2 ) 3 ! x 3 + . . . (1+x)^a=1+ax+\frac{a(a-1)}{2!}x^2+\frac{a(a-1)(a-2)}{3!}x^3+... (1+x)a=1+ax+2!a(a1)x2+3!a(a1)(a2)x3+...

[ ln ⁡ ( x + x 2 ± a 2 ) ] ′ = 1 x 2 ± a 2 . [\ln{(x+\sqrt{x^2\pm a^2})}]'=\frac{1}{\sqrt{x^2\pm a^2}}. [ln(x+x2±a2 )]=x2±a2 1.

[ sin ⁡ ( a x + b ) ] ( n ) = a n sin ⁡ ( a x + b + n π 2 ) . [\sin(ax+b)]^{(n)}=a^n\sin(ax+b+\frac{n\pi}{2}). [sin(ax+b)](n)=ansin(ax+b+2).

[ cos ⁡ ( a x + b ) ] ( n ) = a n cos ⁡ ( a x + b + n π 2 ) . [\cos(ax+b)]^{(n)}=a^n\cos(ax+b+\frac{n\pi}{2}). [cos(ax+b)](n)=ancos(ax+b+2).

[ ln ⁡ ( a x + b ) ] ( n ) = ( − 1 ) n − 1 a n ( n − 1 ) ! ( a x + b ) n . [\ln(ax+b)]^{(n)}=(-1)^{n-1}a^n\frac{(n-1)!}{(ax+b)^n}. [ln(ax+b)](n)=(1)n1an(ax+b)n(n1)!.

( 1 a x + b ) ( n ) = ( − 1 ) n a n n ! ( a x + b ) n + 1 . (\frac{1}{ax+b})^{(n)}=(-1)^n a^n\frac{n!}{(ax+b)^{n+1}}. (ax+b1)(n)=(1)nan(ax+b)n+1n!.

d 2 y d x 2 = d y d t / d t d x / d t . \frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\frac{\frac{\mathrm{d}y}{\mathrm{d}t}/\mathrm{d}t}{\mathrm{d}x/\mathrm{d}t}. dx2d2y=dx/dtdtdy/dt.

K = ∣ d θ d S ∣ = ∣ d arctan ⁡ d y d x ∣ ( d x ) 2 + ( d y ) 2 = ∣ d arctan ⁡ d y d x d x ∣ ⋅ ∣ d x ∣ ( d x ) 2 + ( d y ) 2 = ∣ d 2 y d x ∣ [ ( d x ) 2 + ( d y ) 2 ] 3 2 = ∣ y ′ ′ ∣ [ 1 + ( y ′ ) 2 ] 3 2 . K=|\frac{\mathrm{d}\theta}{\mathrm{d}S}|=\frac{|\mathrm{d}\arctan\frac{\mathrm{d}y}{\mathrm{d}x}|}{\sqrt{(\mathrm{d}x)^2+(\mathrm{d}y)^2}}=\Big|\frac{\mathrm{d}\arctan\frac{\mathrm{d}y}{\mathrm{d}x}}{\mathrm{d}x}\Big|\cdot\frac{|\mathrm{d}x|}{\sqrt{(\mathrm{d}x)^2+(\mathrm{d}y)^2}}=\frac{|\mathrm{d}^2y\ \mathrm{d}x|}{[(\mathrm{d}x)^2+(\mathrm{d}y)^2]^\frac{3}{2}}=\frac{|y''|}{[1+(y')^2]^\frac{3}{2}}. K=dSdθ=(dx)2+(dy)2 darctandxdy= dxdarctandxdy (dx)2+(dy)2 dx=[(dx)2+(dy)2]23d2y dx=[1+(y)2]23y′′.

多项式函数 f ( x ) = ( x − a ) n g ( x ) f(x)=(x-a)^ng(x) f(x)=(xa)ng(x) g ( a ) ≠ 0 g(a)\ne 0 g(a)=0, n > 1 n>1 n>1: n n n 为偶数时 x = a x=a x=a 为极值点; 奇数时为拐点.

f ( x ) = ∏ i = 1 s ( x − a i ) n i f(x)=\prod_{i=1}^s(x-a_i)^{n_i} f(x)=i=1s(xai)ni n i ∈ N + n_i\in\mathbb{N}_+ niN+, a i ≠ a j ( i ≠ j ) a_i\ne a_j\ (i\ne j) ai=aj (i=j): k 1 k_1 k1 n i = 1 n_i=1 ni=1 个数, k 2 k_2 k2 n i > 1 n_i>1 ni>1 为偶数个数, k 3 k_3 k3 n i > 1 n_i>1 ni>1 为奇数个数 ⟹ \implies 极值点个数为 k 1 + 2 k 2 + k 3 − 1 k_1+2k_2+k_3-1 k1+2k2+k31; 拐点个数为 k 1 + 2 k 2 + 3 k 3 − 2 k_1+2k2+3k_3-2 k1+2k2+3k32.

f ( n ) ( x ) = 0 f^{(n)}(x)=0 f(n)(x)=0 k k k 个根 ⟹ f ( x ) = 0 \implies f(x)=0 f(x)=0 至多有 n + k n+k n+k 个根.

可导点不可同时为极值点和拐点.

lim ⁡ n → ∞ 1 n ∑ i = 1 n f ( i n ) = ∫ 0 1 f ( x ) d x . \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^nf(\frac{i}{n})=\int_0^1f(x)\mathrm{d}x. nlimn1i=1nf(ni)=01f(x)dx.

∫ tan ⁡ x d x = − ln ⁡ ∣ cos ⁡ x ∣ + C . \int\tan x\mathrm{d}x=-\ln{|\cos x|}+C. tanxdx=lncosx+C.

∫ cot ⁡ x d x = ln ⁡ ∣ sin ⁡ x ∣ + C . \int\cot x\mathrm{d}x=\ln{|\sin x|}+C. cotxdx=lnsinx+C.

∫ sec ⁡ x d x = ln ⁡ ∣ sec ⁡ x + tan ⁡ x ∣ + C . \int\sec x\mathrm{d}x=\ln{|\sec x+\tan x|}+C. secxdx=lnsecx+tanx+C.

∫ csc ⁡ x d x = ln ⁡ ∣ csc ⁡ x − cot ⁡ x ∣ + C . \int\csc x\mathrm{d}x=\ln{|\csc x-\cot x|}+C. cscxdx=lncscxcotx+C.

∫ 1 a 2 + x 2 d x = 1 ∣ a ∣ arctan ⁡ x ∣ a ∣ + C . \int\frac{1}{a^2+x^2}\mathrm{d}x=\frac{1}{|a|}\arctan\frac{x}{|a|}+C. a2+x21dx=a1arctanax+C.

∫ 1 a 2 − x 2 d x = arcsin ⁡ x ∣ a ∣ + C . \int\frac{1}{\sqrt{a^2-x^2}}\mathrm{d}x=\arcsin\frac{x}{|a|}+C. a2x2 1dx=arcsinax+C.

∫ a 2 − x 2 d x = a 2 2 arcsin ⁡ x ∣ a ∣ + x 2 a 2 − x 2 + C . \int\sqrt{a^2-x^2}\mathrm{d}x=\frac{a^2}{2}\arcsin\frac{x}{|a|}+\frac{x}{2}\sqrt{a^2-x^2}+C. a2x2 dx=2a2arcsinax+2xa2x2 +C.

∫ 1 x 2 ± a 2 d x = ln ⁡ ( x + x 2 ± a 2 ) + C . \int\frac{1}{\sqrt{x^2\pm a^2}}\mathrm{d}x=\ln(x+\sqrt{x^2\pm a^2})+C. x2±a2 1dx=ln(x+x2±a2 )+C.

∫ x 2 ± a 2 d x = x 2 x 2 ± a 2 ± a 2 2 ln ⁡ ( x + x 2 ± a 2 ) + C . \int\sqrt{x^2\pm a^2}\mathrm{d}x=\frac{x}{2}\sqrt{x^2\pm a^2}\pm\frac{a^2}{2}\ln(x+\sqrt{x^2\pm a^2})+C. x2±a2 dx=2xx2±a2 ±2a2ln(x+x2±a2 )+C.

∫ 1 ( x 2 + a 2 ) 2 d x = 1 2 a 2 ∫ [ a 2 − x 2 ( x 2 + a 2 ) 2 + 1 x 2 + a 2 ] d x = 1 2 a 2 ( x x 2 + a 2 + 1 ∣ a ∣ arctan ⁡ x ∣ a ∣ ) . \int\frac{1}{(x^2+a^2)^2}\mathrm{d}x=\frac{1}{2a^2}\int[\frac{a^2-x^2}{(x^2+a^2)^2}+\frac{1}{x^2+a^2}]\mathrm{d}x=\frac{1}{2a^2}(\frac{x}{x^2+a^2}+\frac{1}{|a|}\arctan\frac{x}{|a|}). (x2+a2)21dx=2a21[(x2+a2)2a2x2+x2+a21]dx=2a21(x2+a2x+a1arctanax).

t = tan ⁡ x 2 ; tan ⁡ x = 2 t 1 − t 2 , sin ⁡ x = 2 t 1 + t 2 , cos ⁡ x = 1 − t 2 1 + t 2 ; d x = 2 1 + t 2 d t . t=\tan\frac{x}{2};\ \tan x=\frac{2t}{1-t^2},\ \sin x=\frac{2t}{1+t^2},\ \cos x=\frac{1-t^2}{1+t^2};\ \mathrm{d}x=\frac{2}{1+t^2}\mathrm{d}t. t=tan2x; tanx=1t22t, sinx=1+t22t, cosx=1+t21t2; dx=1+t22dt.

t = tan ⁡ x ; sin ⁡ 2 x = t 1 + t 2 , cos ⁡ 2 x = 1 1 + t 2 ; d x = 1 1 + t 2 d t . t=\tan x;\ \sin^2x=\frac{t}{1+t^2},\ \cos^2x=\frac{1}{1+t^2};\ \mathrm{d}x=\frac{1}{1+t^2}\mathrm{d}t. t=tanx; sin2x=1+t2t, cos2x=1+t21; dx=1+t21dt.

∫ f ⋅ f ′ = f 2 . \int f\cdot f'=f^2. ff=f2.

∫ ( f ′ ) 2 + f ′ ′ ⋅ f = f ⋅ f ′ . \int(f')^2+f''\cdot f=f\cdot f'. (f)2+f′′f=ff.

∫ ( f ′ + f ⋅ φ ′ ) ⋅ e φ = f ⋅ e φ . \int (f'+f\cdot\varphi')\cdot e^\varphi=f\cdot e^\varphi. (f+fφ)eφ=feφ.

∫ ( f ′ ⋅ x − f ) / x 2 = f x . \int (f'\cdot x-f)/x^2=\frac{f}{x}. (fxf)/x2=xf.

∫ [ f ′ ′ ⋅ f − ( f ′ ) 2 ] / f 2 = f ′ f . \int [f''\cdot f-(f')^2]/f^2=\frac{f'}{f}. [f′′f(f)2]/f2=ff.

∫ f ′ f = ln ⁡ f . \int \frac{f'}{f}=\ln f. ff=lnf.

∫ u v ( n + 1 ) = ∑ i = 0 n ( − 1 ) i u ( i ) v ( n − i ) + ( − 1 ) n + 1 ∫ u ( n + 1 ) v . \int uv^{(n+1)}=\sum_{i=0}^n(-1)^iu^{(i)}v^{(n-i)}+(-1)^{n+1}\int u^{(n+1)}v. uv(n+1)=i=0n(1)iu(i)v(ni)+(1)n+1u(n+1)v.

∫ e a x sin ⁡ b x d x = 1 a 2 ( e a x ) ′ sin ⁡ b x − 1 a 2 ( sin ⁡ b x ) ′ e a x − b 2 a 2 ∫ e a x sin ⁡ b x d x . \int e^{ax}\sin{bx}\mathrm{d}x=\frac{1}{a^2}(e^{ax})'\sin{bx}-\frac{1}{a^2}(\sin{bx})'e^{ax}-\frac{b^2}{a^2}\int e^{ax}\sin{bx}\mathrm{d}x. eaxsinbxdx=a21(eax)sinbxa21(sinbx)eaxa2b2eaxsinbxdx.

∫ b a f ( x ) d x = ∫ b a f ( a + b − x ) d x . \int_b^a f(x)\mathrm{d}x=\int_b^a f(a+b-x)\mathrm{d}x. baf(x)dx=baf(a+bx)dx.

∫ − a a a 2 − x 2 d x = π a 2 2 ( a > 0 ) . \int_{-a}^a \sqrt{a^2-x^2}\mathrm{d}x=\frac{\pi a^2}{2}\ (a>0). aaa2x2 dx=2πa2 (a>0).

∫ 0 1 arcsin ⁡ 1 − x 2 d x = 1. \int_0^1 \arcsin\sqrt{1-x^2}\mathrm{d}x=1. 01arcsin1x2 dx=1.

∫ 0 π 2 sin ⁡ n x d x = ∫ 0 π 2 cos ⁡ n x d x = { ( n − 1 ) ! ! n ! ! , n = 2 k + 1 ( n − 1 ) ! ! n ! ! ⋅ π 2 , n = 2 k ( k ∈ N ∗ ) . \int_0^\frac{\pi}{2}\sin^nx\mathrm{d}x=\int_0^\frac{\pi}{2}\cos^nx\mathrm{d}x= \begin{cases} \frac{(n-1)!!}{n!!}, & n=2k+1 \\ \frac{(n-1)!!}{n!!}\cdot\frac{\pi}{2}, & n=2k \end{cases}(k\in\mathbb{N}^*). 02πsinnxdx=02πcosnxdx={n!!(n1)!!,n!!(n1)!!2π,n=2k+1n=2k(kN).

d d x ∫ φ 1 ( x ) φ 2 ( x ) f ( t ) d t = f [ φ 1 ( x ) ] φ 1 ′ ( x ) − f [ φ 2 ( x ) ] φ 2 ′ ( x ) . \frac{\mathrm{d}}{\mathrm{d}x}\int_{\varphi_1(x)}^{\varphi_2(x)}f(t)\mathrm{d}t=f[\varphi_1(x)]\varphi_1'(x)-f[\varphi_2(x)]\varphi_2'(x). dxdφ1(x)φ2(x)f(t)dt=f[φ1(x)]φ1(x)f[φ2(x)]φ2(x).

d d x ∫ 0 x t f ( x − t ) d t = d d x ∫ 0 x ( x − t ) f ( t ) d t = ∫ 0 x f ( t ) d t \frac{\mathrm{d}}{\mathrm{d}x}\int_0^xtf(x-t)\mathrm{d}t=\frac{\mathrm{d}}{\mathrm{d}x}\int_0^x(x-t)f(t)\mathrm{d}t=\int_0^xf(t)\mathrm{d}t dxd0xtf(xt)dt=dxd0x(xt)f(t)dt=0xf(t)dt

变限积分必连续; 被积函数跳跃间断点处不可导, 可去间断点处可导.

∫ 0 1 1 x p \int_0^1\frac{1}{x^p} 01xp1 收敛时 0 < p < 1 0<p<1 0<p<1; 发散时 p ≥ 1 p\geq 1 p1.
∫ 1 + ∞ 1 x p \int_1^{+\infty}\frac{1}{x^p} 1+xp1 收敛时 p > 1 p>1 p>1; 发散时 p ≤ 1 p\leq 1 p1.
∑ n = 1 ∞ 1 n p \sum_{n=1}^\infty\frac{1}{n^p} n=1np1 收敛时 p > 1 p>1 p>1; 发散时 p ≤ 1 p\leq 1 p1.
∑ n = 1 ∞ ( − 1 ) n − 1 1 n p \sum_{n=1}^\infty(-1)^{n-1}\frac{1}{n^p} n=1(1)n1np1 绝对收敛时 p > 1 p>1 p>1; 条件收敛时 0 < p ≤ 1 0<p\leq 1 0<p1.

恒正函数 lim ⁡ x → B f ( x ) g ( x ) = λ \lim_{x\to\mathscr{B}}\frac{f(x)}{g(x)}=\lambda limxBg(x)f(x)=λ: ( ⟹ \implies 比较判敛法极限形式)
λ ≠ 0 , ∞ \lambda\ne 0,\infty λ=0, 时, ∃ ϵ > 0 \exists\epsilon>0 ϵ>0, ( λ − ϵ ) g ( x ) ≤ f ( x ) ≤ ( λ − ϵ ) g ( x ) (\lambda-\epsilon)g(x)\leq f(x)\leq(\lambda-\epsilon)g(x) (λϵ)g(x)f(x)(λϵ)g(x).
λ = 0 \lambda=0 λ=0 时, ∃ ϵ > 0 \exists\epsilon>0 ϵ>0, f ( x ) ≤ ϵ g ( x ) f(x)\leq\epsilon g(x) f(x)ϵg(x).

Γ ( a ) = ∫ 0 + ∞ x a − 1 e − x d x = 2 ∫ 0 + ∞ x 2 a − 1 e − x 2 d x ; Γ ( a ) = ( a − 1 ) Γ ( a − 1 ) ; Γ ( n ) = ( n − 1 ) ! ( n ∈ N + ) ; ∫ − ∞ + ∞ e − x 2 d x ≡ Γ ( 1 2 ) = π . \Gamma(a)=\int_0^{+\infty}x^{a-1}e^{-x}\mathrm{d}x=2\int_0^{+\infty}x^{2a-1}e^{-x^2}\mathrm{d}x;\\ \Gamma(a)=(a-1)\Gamma(a-1);\ \Gamma(n)=(n-1)!\ (n\in\mathbb{N}_+);\ \int_{-\infty}^{+\infty}e^{-x^2}\mathrm{d}x\equiv\Gamma(\frac{1}{2})=\sqrt{\pi}. Γ(a)=0+xa1exdx=20+x2a1ex2dx;Γ(a)=(a1)Γ(a1); Γ(n)=(n1)! (nN+); +ex2dxΓ(21)=π .

平面曲线 y = f ( x ) , x ∈ [ a , b ] y=f(x),x\in[a,b] y=f(x),x[a,b] 绕直线 A x + B y + C = 0 Ax+By+C=0 Ax+By+C=0 旋转得到的体积:
V = π ∫ L r ( s ) 2 h ( s ) d l = π ∫ a b ( ∣ A x + B f ( x ) + C ∣ A 2 + B 2 ) 2 ∣ 1 − A B f ′ ( x ) ∣ 1 1 + A 2 B 2 d x = π ( A 2 + B 2 ) 3 2 ∫ a b [ A x + B f ( x ) + C ] 2 ∣ A f ′ ( x ) − B ∣ d x . V=\pi\int_L r(s)^2h(s)\mathrm{d}l=\pi\int_a^b(\frac{|Ax+Bf(x)+C|}{\sqrt{A^2+B^2}})^2\Big|1-\frac{A}{B}f'(x)\Big|\frac{1}{\sqrt{1+\frac{A^2}{B^2}}}\mathrm{d}x\\ =\frac{\pi}{(A^2+B^2)^\frac{3}{2}}\int_a^b[Ax+Bf(x)+C]^2|Af'(x)-B|\mathrm{d}x. V=πLr(s)2h(s)dl=πab(A2+B2 Ax+Bf(x)+C)2 1BAf(x) 1+B2A2 1dx=(A2+B2)23πab[Ax+Bf(x)+C]2Af(x)Bdx.

微分方程通解含有的独立常数个数等于微分方程的阶数.

y ′ + P ( x ) y = Q ( x ) ⟹ [ e ∫ P ( x ) d x y ] ′ = Q ( x ) e ∫ P ( x ) d x ⟹ y = e − ∫ P ( x ) d x [ ∫ Q ( x ) e ∫ P ( x ) d x d x + C ] . y'+P(x)y=Q(x)\implies [e^{\int P(x)\mathrm{d}x}y]'=Q(x)e^{\int P(x)\mathrm{d}x}\\ \implies y=e^{-\int P(x)\mathrm{d}x}[\int Q(x)e^{\int P(x)\mathrm{d}x}\mathrm{d}x+C]. y+P(x)y=Q(x)[eP(x)dxy]=Q(x)eP(x)dxy=eP(x)dx[Q(x)eP(x)dxdx+C].

y ′ + P ( x ) y = Q ( x ) y n ⟹ y − n y ′ + P ( x ) y − n + 1 = Q ( x ) ⟹ 1 1 − n z x ′ + P ( x ) z = Q ( x ) , z = y − n + 1 . y'+P(x)y=Q(x)y^n\implies y^{-n}y'+P(x)y^{-n+1}=Q(x)\implies\\ \frac{1}{1-n}z_x'+P(x)z=Q(x),\ z=y^{-n+1}. y+P(x)y=Q(x)ynyny+P(x)yn+1=Q(x)1n1zx+P(x)z=Q(x), z=yn+1.

y ′ ′ = f ( y , y ′ ) ⟹ p y ′ p = f ( y , p ) , p = y ′ . y''=f(y,y')\implies p_y'p=f(y,p),\ p=y'. y′′=f(y,y)pyp=f(y,p), p=y.

x 2 y ′ ′ + p x y ′ + q y = f ( x ) ⟹ y t ′ ′ + ( p − 1 ) y t ′ + q y = f ( e t ) , x = e t > 0. x^2y''+pxy'+qy=f(x)\implies y_t''+(p-1)y_t'+qy=f(e^t),\ x=e^t>0. x2y′′+pxy+qy=f(x)yt′′+(p1)yt+qy=f(et), x=et>0.

y ′ ′ + p y ′ + q y = 0 y''+py'+qy=0 y′′+py+qy=0, 特征方程 λ 2 + p λ + q = 0 \lambda^2+p\lambda+q=0 λ2+pλ+q=0.
有不等实根时 y = C 1 e λ 1 x + C 2 e λ 2 x y=C_1e^{\lambda_1 x}+C_2e^{\lambda_2 x} y=C1eλ1x+C2eλ2x;
有相等实根时 y = ( C 1 + C 2 x ) e λ x y=(C_1+C_2x)e^{\lambda x} y=(C1+C2x)eλx;
仅有复根 α ± β i \alpha\pm\beta i α±βi y = e α x ( C 1 cos ⁡ β x + C 2 sin ⁡ β x ) y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x) y=eαx(C1cosβx+C2sinβx).

y ( n ) + ∑ i = 1 n − 1 p i y ( n − i ) + p n y = 0 y^{(n)}+\sum_{i=1}^{n-1}p_iy^{(n-i)}+p_ny=0 y(n)+i=1n1piy(ni)+pny=0, 特征方程 λ n + ∑ i = 1 n − 1 p i λ n − i + p n = 0 \lambda^n+\sum_{i=1}^{n-1}p_i\lambda^{n-i}+p_n=0 λn+i=1n1piλni+pn=0.
有单实根时 y = C e λ x y=Ce^{\lambda x} y=Ceλx;
k k k 重实根时 y = ( ∑ i = 1 k C i x i − 1 ) e λ x y=(\sum_{i=1}^kC_i x^{i-1})e^{\lambda x} y=(i=1kCixi1)eλx;
有一对单复根 α ± β i \alpha\pm\beta i α±βi y = e α x ( C 1 cos ⁡ β x + C 2 sin ⁡ β x ) y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x) y=eαx(C1cosβx+C2sinβx);
有一对 k k k 重复根 α ± β i \alpha\pm\beta i α±βi y = e α x ( C 1 cos ⁡ β x + C 2 sin ⁡ β x + C 3 x cos ⁡ β x + C 4 x sin ⁡ β x ) y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x+C_3x\cos\beta x+C_4x\sin\beta x) y=eαx(C1cosβx+C2sinβx+C3xcosβx+C4xsinβx).

y ′ ′ + p y ′ + q y = f ( x ) y''+py'+qy=f(x) y′′+py+qy=f(x).
f ( x ) = e α x P n ( x ) f(x)=e^{\alpha x}P_n(x) f(x)=eαxPn(x) y ∗ = e α x Q n ( x ) x k y^*=e^{\alpha x}Q_n(x)x^k y=eαxQn(x)xk: α \alpha α 不为特征根时 k = 0 k=0 k=0; 单特征根时 k = 1 k=1 k=1; 二重特征根时 k = 2 k=2 k=2.
f ( x ) = e α x [ P m ( x ) cos ⁡ β x + P n ( x ) sin ⁡ β x ] f(x)=e^{\alpha x}[P_m(x)\cos\beta x+P_n(x)\sin\beta x] f(x)=eαx[Pm(x)cosβx+Pn(x)sinβx] y ∗ = e α x [ Q l ( x ) cos ⁡ β x + R l ( x ) sin ⁡ β x ] x k y^*=e^{\alpha x}[Q_l(x)\cos\beta x+R_l(x)\sin\beta x]x^k y=eαx[Ql(x)cosβx+Rl(x)sinβx]xk, l = max ⁡ { m , n } l=\max\{m,n\} l=max{m,n}: α ± β i \alpha\pm\beta i α±βi 不为特征根时 k = 0 k=0 k=0; 特征根时 k = 1 k=1 k=1.

{ u n } \{u_n\} {un} 单调不增, lim ⁡ n → ∞ u n = 0 ⟹ \lim_{n\to\infty}u_n=0\implies limnun=0 交错级数 ∑ n = 1 ∞ ( − 1 ) n − 1 u n \sum_{n=1}^\infty(-1)^{n-1}u_n n=1(1)n1un 收敛 (莱布尼茨判敛法).

绝对收敛: ∑ n = 1 ∞ ∣ u n ∣ \sum_{n=1}^\infty|u_n| n=1un 收敛时 ( ⟹ ∑ n = 1 ∞ u n \implies \sum_{n=1}^\infty u_n n=1un 收敛).
条件收敛: ∑ n = 1 ∞ u n \sum_{n=1}^\infty u_n n=1un 收敛且 ∑ n = 1 ∞ ∣ u n ∣ \sum_{n=1}^\infty|u_n| n=1un 发散.

函数项级数 ∑ n = 0 ∞ u n ( x ) \sum_{n=0}^\infty u_n(x) n=0un(x) 收敛域 (阿贝尔定理): lim ⁡ n → ∞ ∣ u n + 1 ( x ) ∣ ∣ u n ( x ) ∣ < 1 \lim_{n\to\infty}\frac{|u_{n+1}(x)|}{|u_n(x)|}<1 limnun(x)un+1(x)<1 (达朗贝尔判敛法) 或 lim ⁡ n → ∞ ∣ u n ( x ) ∣ n < 1 \lim_{n\to\infty}\sqrt[n]{|u_n(x)|}<1 limnnun(x) <1 (柯西判敛法) 给出收敛区间 ( a , b ) (a,b) (a,b) 并单独判断 a a a b b b 处敛散性.

幂级数 ∑ n = 0 ∞ a n x n ⋅ ∑ n = 0 ∞ b n x n = ∑ n = 0 ∞ ( ∑ i = 1 n a i b n − i ) x n \sum_{n=0}^\infty a_nx^n\cdot\sum_{n=0}^\infty b_nx^n=\sum_{n=0}^\infty(\sum_{i=1}^n a_ib_{n-i})x^n n=0anxnn=0bnxn=n=0(i=1naibni)xn, 收敛半径 R ≥ min ⁡ { R a , R b } R\geq\min\{R_a,R_b\} Rmin{Ra,Rb}.

幂级数恒等变换: ∑ n = k ∞ a n x n = ∑ n = k + l ∞ a n − l x n − l \sum_{n=k}^\infty a_nx^n=\sum_{n=k+l}^\infty a_{n-l}x^{n-l} n=kanxn=n=k+lanlxnl; ∑ n = k ∞ a n x n = ∑ n = k k + l − 1 a n x n + ∑ n = k + l ∞ a n x n \sum_{n=k}^\infty a_nx^n=\sum_{n=k}^{k+l-1} a_nx^n+\sum_{n=k+l}^\infty a_nx^n n=kanxn=n=kk+l1anxn+n=k+lanxn; ∑ n = k ∞ a n x n = x l ∑ n = k ∞ a n x n − l \sum_{n=k}^\infty a_nx^n=x^l\sum_{n=k}^\infty a_nx^{n-l} n=kanxn=xln=kanxnl.

幂级数的和函数 S ( x ) = ∑ n = 0 ∞ a n x n S(x)=\sum_{n=0}^\infty a_nx^n S(x)=n=0anxn 在收敛域上连续.
幂级数及和函数在收敛域上可积(可导)时, 有逐项积分(求导): ∫ 0 x S ( x ) d x = ∑ n = 0 ∞ a n n + 1 x n + 1 \int_0^x S(x)\mathrm{d}x=\sum_{n=0}^\infty \frac{a_n}{n+1}x^{n+1} 0xS(x)dx=n=0n+1anxn+1; S ′ ( x ) = ∑ n = 1 ∞ n a n x n − 1 S'(x)=\sum_{n=1}^\infty na_nx^{n-1} S(x)=n=1nanxn1.

∑ n = 1 ∞ x n n = ∫ 0 x ( ∑ n = 0 ∞ x n ) d x = − ln ⁡ ( 1 − x ) , x ∈ [ − 1 , 1 ) . \sum_{n=1}^\infty\frac{x^n}{n}=\int_0^x(\sum_{n=0}^\infty x^n)\mathrm{d}x=-\ln(1-x),\ x\in[-1,1). n=1nxn=0x(n=0xn)dx=ln(1x), x[1,1).

∑ n = 1 ∞ n x n − 1 = ( ∑ n = 1 ∞ x n ) ′ = 1 ( 1 − x ) 2 , x ∈ ( − 1 , 1 ) . \sum_{n=1}^\infty nx^{n-1}=(\sum_{n=1}^\infty x^n)'=\frac{1}{(1-x)^2},\ x\in(-1,1). n=1nxn1=(n=1xn)=(1x)21, x(1,1).

1 1 + x = ∑ n = 0 ∞ ( − 1 ) n x n , x ∈ ( − 1 , 1 ) . \frac{1}{1+x}=\sum_{n=0}^\infty(-1)^nx^n,\ x\in(-1,1). 1+x1=n=0(1)nxn, x(1,1).

e x = ∑ n = 0 ∞ x n n ! , x ∈ R . e^x=\sum_{n=0}^\infty\frac{x^n}{n!},\ x\in\mathbb{R}. ex=n=0n!xn, xR.

sin ⁡ x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! , x ∈ R . \sin x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!},\ x\in\mathbb{R}. sinx=n=0(1)n(2n+1)!x2n+1, xR.

f ( x + 2 l ) = f ( x ) ⟹ f ( x ) ∼ ( a 0 2 + ∑ n = 1 ∞ a n cos ⁡ n π l x ) + ( ∑ n = 1 ∞ b n sin ⁡ n π l x ) ; a n = 1 l ∫ − l l f ( x ) cos ⁡ n π l x d x , b n = 1 l ∫ − l l f ( x ) sin ⁡ n π l x d x . f(x+2l)=f(x)\implies f(x)\sim (\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos\frac{n\pi}{l}x)+(\sum_{n=1}^\infty b_n\sin\frac{n\pi}{l}x);\\ a_n=\frac{1}{l}\int_{-l}^l f(x)\cos\frac{n\pi}{l}x\mathrm{d}x,\ b_n=\frac{1}{l}\int_{-l}^l f(x)\sin\frac{n\pi}{l}x\mathrm{d}x. f(x+2l)=f(x)f(x)(2a0+n=1ancoslx)+(n=1bnsinlx);an=l1llf(x)coslxdx, bn=l1llf(x)sinlxdx.

f ( x ) f(x) f(x) 2 l 2l 2l 为周期, 在 [ − l , l ] [-l,l] [l,l] 上可积, 至多有有限个第一类间断点及极值点 ⟹ f ( x ) \implies f(x) f(x) 傅里叶级数几乎处处收敛, 和函数 S ( x ) = f ( x ) S(x)=f(x) S(x)=f(x), x x x 处连续; f ( x − 0 ) + f ( x + 0 ) 2 \frac{f(x-0)+f(x+0)}{2} 2f(x0)+f(x+0), x x x 处间断; f ( − l + 0 ) + f ( l − 0 ) 2 \frac{f(-l+0)+f(l-0)}{2} 2f(l+0)+f(l0), x = ± l x=\pm l x=±l.

f ( x ) f(x) f(x) [ 0 , l ] [0,l] [0,l] 上可积:
奇延拓: F ( x ) = f ( x ) , k l < x ≤ ( k + 1 ) l ; − f ( − x ) , ( k − 1 ) l ≤ x < k l ; 0 , x = k l ; k ∈ Z F(x)=f(x),\ kl<x\leq(k+1)l;\ -f(-x),\ (k-1)l\leq x<kl;\ 0, \ x=kl;\ k\in\mathbb{Z} F(x)=f(x), kl<x(k+1)l; f(x), (k1)lx<kl; 0, x=kl; kZ.
偶延拓: F ( x ) = f ( x ) , k l ≤ x ≤ ( k + 1 ) l ; f ( − x ) , ( k − 1 ) l ≤ x < k l ; k ∈ Z F(x)=f(x),\ kl\leq x\leq(k+1)l;\ f(-x),\ (k-1)l\leq x<kl;\ k\in\mathbb{Z} F(x)=f(x), klx(k+1)l; f(x), (k1)lx<kl; kZ.

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