微积分A(1)期末复习笔记

微积分A(1)期末复习笔记(仅包括下半学期相关内容),仅供参考。 PDF版本链接

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定理

黎曼积分

abf(x)  dx=limλ(P)0σ(f;P,ξ)=limλ(P)0i=1nf(ξi)Δxi\int_a^b f(x) \;\text{d}x = \lim_{\lambda(P) \to 0} \sigma(f;P,\xi) = \lim_{\lambda(P) \to 0} \sum_{i=1}^n f(\xi_i) \Delta x_i


ff可积:

IFF ε>0,U(f;P)L(f;P)<ε\forall \varepsilon \gt 0, U(f;P)-L(f;P) \lt \varepsilon.

IFF abf(x)  dx=abf(x)  dx\underline{\int}_a^b f(x) \;\text{d}x = \overline{\int}_a^b f(x) \;\text{d}x.

IFF limλ(P)0i=1nω(f;Δi)Δxi=0\lim_{\lambda(P) \to 0} \sum_{i=1}^n \omega(f;\Delta_i) \Delta x_i = 0.


一致连续

ε>0,δ>0,x,yX,xy<δ\forall \varepsilon \gt 0, \exists \delta \gt 0, \forall x,y \in X, \vert x-y \vert \lt \delta, 都有f(x)f(y)<ε\vert f(x) - f(y) \vert \lt \varepsilon, 则ffXX上一致连续.


ff一致连续:

IFF {xn},{yn},limn(xnyn)=0\forall \lbrace x_n \rbrace, \lbrace y_n \rbrace, \lim_{n \to \infty} (x_n-y_n) = 0, 都有limn(f(xn)f(yn))=0\lim_{n \to \infty} (f(x_n)-f(y_n)) = 0.


LL-Lipschitz函数一致连续.

闭区间上连续函数一致连续.

闭区间上连续函数可积.

闭区间上单调函数可积.

有界函数可积当且仅当的间断点集零测度.


积分中值定理

积分第一中值定理: fC[a,b],ξ[a,b],abf(x)  dx=f(ξ)(ba)f \in \mathscr C[a,b], \exists \xi \in [a,b], \int_a^b f(x) \;\text{d}x = f(\xi)(b-a).

加强积分第一中值定理: fR[a,b],fC(a,b),ξ(a,b),abf(x)  dx=f(ξ)(ba)f \in \mathscr R[a,b], f \in \mathscr C(a,b), \exists \xi \in (a,b), \int_a^b f(x) \;\text{d}x = f(\xi)(b-a).

广义积分第一中值定理: fC[a,b],gR[a,b],ξ[a,b],abf(x)g(x)  dx=f(ξ)abg(x)  dxf \in \mathscr C[a,b], g \in \mathscr R[a,b], \exists \xi \in [a,b], \int_a^b f(x)g(x) \;\text{d}x = f(\xi)\int_a^b g(x) \;\text{d}x.

积分第二中值定理: fR[a,b]f \in \mathscr R[a,b], gg[a,b][a,b]上单调, ξ[a,b],abf(x)g(x)  dx=g(a)aξf(x)  dx+g(b)ξbf(x)  dx\exists \xi \in [a,b], \int_a^b f(x)g(x) \;\text{d}x = g(a) \int_a^\xi f(x) \;\text{d}x + g(b)\int_\xi^b f(x) \;\text{d}x.


广义积分收敛性判断

Cauchy判别准则: aωf(x)  dx\int_a^\omega f(x) \;\text{d}x收敛:

IFF ε>0,c(a,ω),A1,A2(c,ω),A1A2f(x)  dx<ε\forall \varepsilon \gt 0, \exists c \in (a,\omega), \forall A_1,A_2 \in (c,\omega), \vert \int_{A_1}^{A_2} f(x) \;\text{d}x \vert \lt \varepsilon.

Abel判别准则: aωf(x)  dx\int_a^\omega f(x) \;\text{d}x收敛且gg单调有界则aωf(x)g(x)  dx\int_a^\omega f(x)g(x) \;\text{d}x收敛.

Dirichlet判别准则: F(A)=aAf(x)  dxF(A) = \int_a^A f(x) \;\text{d}x有界且gg单调趋于00, 则aωf(x)g(x)  dx\int_a^\omega f(x)g(x) \;\text{d}x收敛.

比较判敛法.

测试函数.


公式

高阶导数公式
  1. (xα)(n)=αnxαn(x^\alpha)^{(n)} = \alpha^{\underline{n}}x^{\alpha-n}.

  2. (eαx)(n)=αneαx(e^{\alpha x})^{(n)} = \alpha^n e^{\alpha x} (αC\alpha \in \mathbb C).

  3. (ln(1+x))(n)=(1)n1(n1)!(1+x)n(\ln(1+x))^{(n)} = \frac{(-1)^{n-1}(n-1)!}{(1+x)^n}.

  4. sin(n)(x)=sin(x+nπ2)\sin^{(n)}(x) = \sin(x+\frac{n\pi}{2}), cos(n)(x)=cos(x+nπ2)\cos^{(n)}(x) = \cos(x+\frac{n\pi}{2}).

131 \Rightarrow 3:

(ln(1+x))=11+x=(1+x)1[(ln(1+x))](n1)=1n1(1+x)1(n1)=(1)n1(n1)!(1+x)n(\ln(1+x))' = \frac{1}{1+x} = (1+x)^{-1} \\ [(\ln(1+x))']^{(n-1)} = -1^{\underline{n-1}}(1+x)^{-1-(n-1)} = \frac{(-1)^{n-1}(n-1)!}{(1+x)^n} \\

242 \Rightarrow 4:

cos(n)(x)+isin(n)(x)=(cosx+isinx)(n)=(eix)(n)=ineix=(eiπ2)neix=ei(x+nπ2)=cos(x+nπ2)+isin(x+nπ2)\cos^{(n)}(x) + i\sin^{(n)}(x) = (\cos x + i\sin x)^{(n)} = (e^{ix})^{(n)} = \\ i^ne^{ix} = (e^\frac{i\pi}{2})^ne^{ix} = e^{i(x+\frac{n\pi}{2})} = \cos(x+\frac{n\pi}{2}) + i\sin(x+\frac{n\pi}{2}) \\

(λf+μg)(n)=λf(n)+μg(n)(fg)(n)=k=0n(nk)f(k)g(nk)(\lambda f + \mu g)^{(n)} = \lambda f^{(n)} + \mu g^{(n)} \\ (fg)^{(n)} = \sum_{k=0}^n {n \choose k} f^{(k)}g^{(n-k)} \\


常用泰勒展开

ex=1+x+x22+x36+x424+...sinx=xx36+x5120...cosx=1x22+x424...ln(1+x)=xx2+x33x44+...ln(11x)=x+x2+x33+x44+...11x=1+x+x2+x3+x4+...\begin{aligned} e^x &= 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + ... \\ \sin x &= x - \frac{x^3}{6} + \frac{x^5}{120} - ... \\ \cos x &= 1 - \frac{x^2}{2} + \frac{x^4}{24} - ... \\ \ln(1+x) &= x - \frac{x}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ... \\ \ln(\frac{1}{1-x}) &= x + \frac{x}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ... \\ \frac{1}{1-x} &= 1 + x + x^2 + x^3 + x^4 + ... \\ \end{aligned}


变限积分求导

(ψ(x)φ(x)f(t)  dt)=f(φ(x))φ(x)f(ψ(x))ψ(x)\left( \int_{\psi(x)}^{\varphi(x)} f(t) \;\text{d}t \right)' = f(\varphi(x)) \varphi'(x) - f(\psi(x))\psi'(x)

(ψ(x)φ(x)f(x,t)  dt)=f(x,φ(x))φ(x)f(x,ψ(x))ψ(x)+ψ(x)φ(x)df(x,t)dx  dt\left( \int_{\psi(x)}^{\varphi(x)} f(x,t) \;\text{d}t \right)' = f(x,\varphi(x))\varphi'(x) - f(x,\psi(x))\psi'(x) + \int_{\psi(x)}^{\varphi(x)} \frac{\text{d}f(x,t)}{\text{d}x} \;\text{d}t


定积分与数列极限

limnbank=1nf(ξn,k)=abf(x)  dx\lim_{n \to \infty}\frac{b-a}{n}\sum_{k=1}^n f(\xi_{n,k}) = \int_a^b f(x) \;\text{d}x


不等式

均值不等式:

ni=1n1xii=1nxin1ni=1nxii=1nxi2n\frac{n}{\sum_{i=1}^n\frac{1}{x_i}} \le \sqrt[n]{\prod_{i=1}^nx_i} \le \frac{1}{n} \sum_{i=1}^n x_i \le \sqrt{\frac{\sum_{i=1}^n x_i^2}{n}}

Young不等式(1p+1q=1\frac{1}{p}+\frac{1}{q} = 1):

x1py1q1px+1qyx^\frac{1}{p}y^{\frac{1}{q}} \le \frac{1}{p}x + \frac{1}{q}y

Holder不等式(1p+1q=1\frac{1}{p}+\frac{1}{q} = 1):

k=1nxkyk(k=1nxkp)1p(k=1nykq)1q\sum_{k=1}^n x_ky_k \le \left( \sum_{k=1}^nx_k^p \right)^{\frac{1}{p}} \left( \sum_{k=1}^ny_k^q \right)^{\frac{1}{q}}

积分Cauchy不等式:

(abf(x)g(x)  dx)2(abf2(x)  dx)(abg2(x)  dx)\left( \int_a^b f(x)g(x) \;\text{d}x \right)^2 \le \left( \int_a^b f^2(x) \;\text{d}x \right) \left( \int_a^b g^2(x) \;\text{d}x \right)

积分Holder不等式(1p+1q=1\frac{1}{p}+\frac{1}{q} = 1):

abf(x)g(x)  dx(abf(x)p  dx)1p(abf(x)q  dx)1q\left\vert \int_a^b f(x)g(x) \;\text{d}x \right\vert \le \left( \int_a^b \vert f(x) \vert^p \;\text{d}x \right)^\frac{1}{p} \left( \int_a^b \vert f(x) \vert^q \;\text{d}x \right)^\frac{1}{q}

积分Jensen不等式(φ\varphi为凸函数):

φ(1baabf(x)  dx)1baabφ(f(x))  dx\varphi \left( \frac{1}{b-a} \int_a^b f(x) \;\text{d}x \right) \le \frac{1}{b-a} \int_a^b \varphi(f(x)) \;\text{d}x

积分Minkowski不等式(p>1p \gt 1):

(ab(f(x)+g(x))p  dx)1p(abf(x)p  dx)1p+(abg(x)p  dx)1p\left( \int_a^b (\vert f(x) \vert + \vert g(x) \vert)^p \;\text{d}x \right)^{\frac{1}{p}} \le \left( \int_a^b \vert f(x) \vert^p \;\text{d}x \right)^{\frac{1}{p}} + \left( \int_a^b \vert g(x) \vert^p \;\text{d}x \right)^{\frac{1}{p}}


几何图形计算

直角坐标系下面积(矩形逼近):

S=abf(x)  dxS = \int_a^b \vert f(x) \vert \;\text{d}x

极坐标系下面积(扇形逼近):

S=12αβρ2(θ)  dθS = \frac{1}{2} \int_\alpha^\beta \rho^2(\theta) \;\text{d}\theta

参数方程面积:

x(t)x(t)单调, 反函数t(x)t(x)存在, a=x(α)a = x(\alpha), b=x(β)b = x(\beta), 则:

S=aby(t(x))  dx=x=x(t)αβy(t)x(t)  dtS = \int_a^b \vert y(t(x)) \vert \;\text{d}x \stackrel{x=x(t)}{=} \int_\alpha^\beta \vert y(t)x'(t) \vert \;\text{d}t


直角坐标系弧长:

L=ab1+(f(x))2  dtL = \int_a^b \sqrt{1+(f'(x))^2} \;\text{d}t

极坐标系弧长:

L=αβρ2(θ)+ρ2(θ)  dθL = \int_\alpha^\beta \sqrt{\rho^2(\theta) + \rho'^2(\theta)} \;\text{d}\theta

参数方程曲线弧长:

L=ab(x(t))2+(y(t))2  dtL = \int_a^b \sqrt{(x'(t))^2+(y'(t))^2} \;\text{d}t


曲线的曲率: κ=dαdL\kappa = \frac{\text{d}\alpha}{\text{d}L}. 曲线的曲率半径R=1κR = \frac{1}{\kappa}

参数方程曲线曲率:

κ=dαdL=darctany(t)x(t)x2(t)+y2(t)=x(t)y(t)x(t)y(t)(x2(t)+y2(t))32\kappa = \left \vert \frac{\text{d}\alpha}{\text{d}L} \right \vert = \left \vert \frac{\text{d}\arctan\frac{y'(t)}{x'(t)}}{\sqrt{x'^2(t)+y'^2(t)}} \right \vert = \frac{\vert x'(t)y''(t) - x''(t)y'(t) \vert}{(x'^2(t)+y'^2(t))^\frac{3}{2}}

直角坐标系曲线曲率:

κ=y(x)(1+y2(x))32\kappa = \frac{\vert y''(x) \vert}{(1+y'^2(x))^\frac{3}{2}}

极坐标系曲线曲率:

κ=ρ2(θ)+2ρ2(θ)ρ(θ)ρ(θ)(ρ2(θ)+ρ2(θ))32\kappa = \frac{\vert \rho^2(\theta)+2\rho'^2(\theta)-\rho(\theta)\rho''(\theta) \vert}{(\rho^2(\theta)+\rho'^2(\theta))^{\frac{3}{2}}}


y=f(x)y = f(x)xx轴旋转体体积:

V=πabf2(x)  dxV = \pi \int_a^b f^2(x) \;\text{d}x

y=f(x)y = f(x)yy轴旋转体体积:

dVdx(x)=limΔx0(π(x+Δx)2πx2)y(x)Δx=2πxy(x)=2πxf(x)V=2πabxf(x)  dx\frac{\text{d}V}{\text{d}x}(x) = \lim_{\Delta x \to 0}\frac{(\pi(x+\Delta x)^2-\pi x^2)\vert y(x) \vert}{\Delta x} = 2\pi x \vert y(x) \vert = 2\pi x f(x) \\ V = 2\pi \int_a^b \vert xf(x) \vert \;\text{d}x


极坐标曲线与原点连线所成图形绕极轴旋转体积(一般情况下适用):

V=2π3αβr3(θ)sinθ  dθV = \frac{2\pi}{3}\int_\alpha^\beta r^3(\theta) \sin \theta \;\text{d}\theta


y=f(x)y = f(x)xx轴旋转体侧面积微元: dS=2πy  dL\text{d}S = 2\pi \vert y \vert\;\text{d} L.

参数方程旋转体侧面积(xx轴):

S=2πaby(t)x2(y)+y2(t)  dtS = 2 \pi \int_a^b \vert y(t) \vert \sqrt{x'^2(y)+y'^2(t)} \;\text{d}t

参数方程旋转体侧面积(yy轴):

S=2πabx(t)x2(y)+y2(t)  dtS = 2 \pi \int_a^b \vert x(t) \vert \sqrt{x'^2(y)+y'^2(t)} \;\text{d}t

直角坐标系旋转体侧面积(xx轴):

S=2πaby(x)1+y2(x)  dxS = 2 \pi \int_a^b \vert y(x) \vert \sqrt{1+y'^2(x)} \;\text{d}x

直角坐标系旋转体侧面积(yy轴):

S=2πabx1+y2(x)  dxS = 2 \pi \int_a^b \vert x \vert \sqrt{1+y'^2(x)} \;\text{d}x

极坐标系旋转体侧面积(xx轴):

S=2παβρ(θ)sinθρ2(θ)+ρ2(θ)  dθS = 2 \pi \int_\alpha^\beta \vert \rho(\theta)\sin\theta \vert \sqrt{\rho^2(\theta)+\rho'^2(\theta)} \;\text{d}\theta

极坐标系旋转体侧面积(yy轴):

S=2παβρ(θ)cosθρ2(θ)+ρ2(θ)  dθS = 2 \pi \int_\alpha^\beta \vert \rho(\theta)\cos\theta \vert \sqrt{\rho^2(\theta)+\rho'^2(\theta)} \;\text{d}\theta


曲线质心: 力矩平衡. 设线密度μ\mu为常数. 则一小段线段的质量为dm=μdL\text{d} m = \mu \text{d} L. 线段总质量为m=μ  dLm = \int \mu \;\text{d}L.

沿yy轴方向的静力矩为: My=xμ  dLM_y = \int x\mu \;\text{d}L. 类似地, Mx=yμ  dLM_x = \int y\mu \;\text{d}L.

设质心坐标为(xˉ,yˉ)(\bar x,\bar y), 则:

xˉ=Mym=xμ  dLμ  dL=x  dLL\bar x = \frac{M_y}{m} = \frac{\int x \mu \;\text{d}L}{\int \mu \;\text{d}L} = \frac{\int x \;\text{d}L}{L}

类似地,

yˉ=y  dLL\bar y = \frac{\int y \;\text{d}L}{L}

若线密度不为常数,将μ\mu替换为μ(x)\mu(x)即可,此时分子分母积分不再能消去μ(x)\mu(x).


ODE
直接积分型ODE:

dydx=f(x)\frac{\text{d}y}{\text{d}x} = f(x)

则直接积分. 通解为:

y=f(x)  dxy = \int f(x) \;\text{d}x


一阶线性齐次ODE

dydx+P(x)y=0\frac{\text{d}y}{\text{d}x} + P(x)y = 0 \\

求导后能和自身抵消, 猜测函数为Cef(x)Ce^{f(x)}形式, 解出. 通解为:

y=CeP(x)  dxy = C e^{-\int P(x) \;\text{d}x} \\

注意这里的积分符号表示任意一个原函数, 因此积分结果本身不需要+C+C.


一阶线性非齐次ODE:

dydx+P(x)y=Q(x)\frac{\text{d}y}{\text{d}x} + P(x)y = Q(x) \\

常数变易法, 猜测函数为y=C(x)ef(x)y = C(x)e^{f(x)}, 解出. 通解为:

y=eP(x)  dx(C+Q(x)eP(x)  dx  dx)y = e^{-\int P(x) \;\text{d}x} (C + \int Q(x)e^{\int P(x) \;\text{d}x} \;\text{d}x) \\


分离变量型一阶ODE:

dydx=f(x)g(y)\frac{\text{d}y}{\text{d}x} = f(x)g(y) \\

移项: dyg(y)=f(x)  dx\frac{\text{d}y}{g(y)} = f(x) \;\text{d}x, 因此可以直接积分. 通解为(隐函数):

1g(y)  dy=f(x)  dx+C\int \frac{1}{g(y)} \;\text{d}y = \int f(x) \;\text{d}x + C \\

若存在g(y0)=0g(y_0) = 0, 则y0y_0为该ODE的一个奇解.


ax+by+c型ODE

dydx=f(ax+by+c)\frac{\text{d}y}{\text{d}x} = f(ax+by+c) \\

b=0b = 0, 则为直接积分型ODE. 通解为:

y=f(ax+c)  dx+Cy = \int f(ax+c) \;\text{d}x + C \\

u=ax+by+cu = ax+by+c, 则dudx=a+bdydx=a+bf(u)\frac{\text{d}u}{\text{d}x} = a+b \frac{\text{d}y}{\text{d}x} = a+bf(u), 为分离变量型ODE. 通解为(隐函数):

1a+bf(u)  du=x+C\int \frac{1}{a+bf(u)} \;\text{d}u = x + C \\

若存在a+bf(u0)=0a+bf(u_0) = 0, 则y=1b(u0axc)y = \frac{1}{b}(u_0-ax-c)为该ODE的一个奇解.


y/x型ODE

dydx=F(yx)\frac{\text{d}y}{\text{d}x} = F\left(\frac{y}{x}\right)

u=yxu = \frac{y}{x}, 则dydx=duxdx=xdudx+u=F(u)\frac{\text{d}y}{\text{d}x} = \frac{\text{d}ux}{\text{d}x} = x\frac{\text{d}u}{\text{d}x} + u = F(u), 为分离变量型ODE: xdudx=F(u)ux\frac{\text{d}u}{\text{d}x} = F(u)-u. 通解为(隐函数):

1F(u)u  du=lnx+C\int \frac{1}{F(u)-u} \;\text{d}u = \ln \vert x \vert + C \\

若存在F(u0)u0=0F(u_0)-u_0 = 0, 则y=u0xy = u_0x为该ODE的一个奇解.


x/y型ODE

dydx=F(xy)\frac{\text{d}y}{\text{d}x} = F\left(\frac{x}{y}\right)

u=xyu = \frac{x}{y}, 则dydx=1duydy=1ydudy+u=F(u)\frac{\text{d}y}{\text{d}x} = \frac{1}{\frac{\text{d}uy}{\text{d}y}} = \frac{1}{y\frac{\text{d}u}{\text{d}y} + u} = F(u), 为分离变量型ODE: ydudy=1F(u)uy\frac{\text{d}u}{\text{d}y} = \frac{1}{F(u)}-u. 通解为(隐函数):

11F(u)u  du=lny+C\int \frac{1}{\frac{1}{F(u)}-u} \;\text{d}u = \ln \vert y \vert + C \\

若存在1F(u0)u0=0\frac{1}{F(u_0)}-u_0 = 0, 则y=1u0xy = \frac{1}{u_0}x为该ODE的一个奇解.


直线交点型ODE

dydx=f(a1x+b1y+c1a2x+b2y+c2)\frac{\text{d}y}{\text{d}x} = f\left(\frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}\right)

若直线a1x+b1y+c1=0a_1x+b_1y+c_1=0a2x+b2y+c2=0a_2x+b_2y+c_2=0有唯一交点(x0,y0)(x_0,y_0), 则X=xx0,Y=yy0X = x-x_0, Y = y-y_0:

dYdX=f(a1X+b1Ya2X+b2Y)=f(a1+b1YXa2+b2YX)=F(YX)\frac{\text{d}Y}{\text{d}X} = f\left(\frac{a_1X+b_1Y}{a_2X+b_2Y}\right) = f\left(\frac{a_1+b_1\frac{Y}{X}}{a_2+b_2\frac{Y}{X}}\right) = F(\frac{Y}{X})

否则两直线平行, a1b2=a2b1a_1b_2 = a_2b_1. 则:

f(a1x+b1y+c1a2x+b2y+c2)=f(k+c1kc2a2x+b2y+c2)=F(a2x+b2y+c2)f\left(\frac{a_{1} x+b_{1} y+c_{1}}{a_{2} x+b_{2} y+c_{2}}\right) = f\left(k+\frac{c_{1}-k c_{2}}{a_{2} x+b_{2} y+c_{2}}\right) = F\left(a_{2} x+b_{2} y+c_{2}\right)


伯努利方程

dydx+p(x)y=q(x)yα\frac{\text{d}y}{\text{d}x} + p(x)y = q(x)y^\alpha

α=0\alpha = 0则为一阶线性非齐次ODE, 若α=1\alpha = 1则为分离变量型一阶ODE, 否则令z=y1αz = y^{1-\alpha}, 则dzdx=(1α)yαdydx\frac{\text{d}z}{\text{d}x} = (1-\alpha)y^{-\alpha}\frac{\text{d}y}{\text{d}x}, 为一阶线性非齐次ODE: dzdx+(1α)p(x)z=(1α)q(x)\frac{\text{d}z}{\text{d}x} + (1-\alpha)p(x)z = (1-\alpha)q(x). 通解为:

y=(e(1α)p(x)  dx(C+(1α)q(x)e(1α)p(x)  dx  dx))11αy = \left(e^{-(1-\alpha) \int p(x) \;\text{d}x} (C + (1-\alpha)\int q(x)e^{(1-\alpha) \int p(x) \;\text{d}x} \;\text{d}x)\right)^{\frac{1}{1-\alpha}} \\

另外, 若α>0\alpha \gt 0, y=0y = 0为该ODE的奇解.


高阶ODE与基本解组

nn阶ODE: y(n)+i=0n1ai(x)y(i)=f(x)y^{(n)} + \sum_{i=0}^{n-1} a_i(x)y^{(i)} = f(x). 对于任意一个柯西初值问题, 区间II上存在唯一解.

高阶齐次ODE解集为nn维线性空间. 可以找到nn个线性无关的函数作为基底(基本解组).


Wronsky行列式:

W(f1,f2,,fn)(x)=det(f1(x)f2(x)f3(x)fn(x)f1(x)f2(x)f3(x)fn(x)f1(x)f2(x)f3(x)fn(x)f1(n1)(x)f2(n1)(x)f3(n1)(x)fn(n1)(x))W(f_1,f_2,\cdots,f_n)(x) = \\ \det \begin{pmatrix} f_1(x) & f_2(x) & f_3(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & f_3'(x) & \cdots & f_n'(x) \\ f_1''(x) & f_2''(x) & f_3''(x) & \cdots & f_n''(x) \\ & \vdots & & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & f_3^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \\ \end{pmatrix}

THEOREM

ff线性相关 \Leftrightarrow 在区间IIW(f1,f2,,fn)(x)0W(f_1,f_2,\cdots,f_n)(x) \equiv 0. 且ff线性无关 \Leftrightarrow 在区间IIW(f1,f2,,fn)(x)W(f_1,f_2,\cdots,f_n)(x)恒不为00.


给定一个基本解组, 构造以这个基本解组为解的ODE:

det(f1(x)f2(x)f3(x)fn(x)f1(x)f2(x)f3(x)fn(x)f1(x)f2(x)f3(x)fn(x)f1(n1)(x)f2(n1)(x)f3(n1)(x)fn(n1)(x)f1(n)(x)f2(n)(x)f3(n)(x)fn(n)(x)yyyy(n1)y(n))=0\det \left( \begin{matrix} f_1(x) & f_2(x) & f_3(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & f_3'(x) & \cdots & f_n'(x) \\ f_1''(x) & f_2''(x) & f_3''(x) & \cdots & f_n''(x) \\ & \vdots & & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & f_3^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \\ f_1^{(n)}(x) & f_2^{(n)}(x) & f_3^{(n)}(x) & \cdots & f_n^{(n)}(x) \\ \end{matrix} \right\vert \left. \begin{matrix} y \\ y' \\ y'' \\ \vdots \\ y^{(n-1)} \\ y^{(n)} \\ \end{matrix} \right) = 0


高阶积分型ODE

y(n)=f(x)y^{(n)} = f(x)

nn次原函数.


降阶型ODE

y(n)=F(x,y(k),,y(n1))y^{(n)} = F(x,y^{(k)}, \cdots, y^{(n-1)})

其中k1k \ge 1, 则令p(x)=y(k)(x)p(x) = y^{(k)}(x), 阶数降低, 解出p(x)p(x)后为高阶积分型ODE, y(x)y(x)p(x)p(x)kk次原函数.


不显含x型二阶ODE

F(y,dydx,d2ydx2)=0F(y,\frac{\text{d}y}{\text{d}x},\frac{\text{d}^2y}{\text{d}x^2}) = 0

p=dydxp = \frac{\text{d}y}{\text{d}x}, 则d2ydx2=pdpdy\frac{\text{d}^2y}{\text{d}x^2} = p\frac{\text{d}p}{\text{d}y}, 转换为F(y,p,pdpdy)=0F(y,p,p\frac{\text{d}p}{\text{d}y}) = 0, 为分离变量型ODE: dydx=p(y)\frac{\text{d}y}{\text{d}x} = p(y).


二阶线性齐次ODE

y+py+qy=0y'' + py' + qy = 0

求导后能和自身抵消, 猜测函数为eλxe^{\lambda x}形式, 解出(λ2+pλ+q)eλx=0(\lambda^2 + p\lambda +q)e^{\lambda x} = 0. 判别式Δ=p24q\Delta = p^2-4q

Δ>0\Delta \gt 0, 通解为: y=C1eλ1x+C2eλ2xy = C_1e^{\lambda_1 x} + C_2 e^{\lambda_2 x}.

Δ=0\Delta = 0, 通解为: y=(C1+C2x)eλ=(C1+C2x)ep2xy = (C_1+C_2x)e^\lambda = (C_1+C_2x)e^{-\frac{p}{2}x}.

Δ<0\Delta \lt 0, λ1,λ2\lambda_1,\lambda_2可能为复数α±βi\alpha \pm \beta i. 对于一个复值函数解y(x)y(x), u(x)=Re(x)u(x) = \text{Re}(x)v(x)=Im(x)v(x) = \text{Im(x)}分别满足:

u+pu+qu=Re0=0v+pv+qv=Im0=0u'' + pu' + qu = \text{Re} 0 = 0 \\ v'' + pv' + qv = \text{Im} 0 = 0 \\

故复值函数解y=Ce(α+βi)xy = Ce^{(\alpha + \beta i)x}可以构造两个实值函数解:

y1=eαxcos(βx)y2=eαxsin(βx)y_1 = e^{\alpha x} \cos (\beta x) y_2 = e^{\alpha x} \sin (\beta x)

可以检验两个解线性无关, 则实值通解为:

y=eαx(C1cos(βx)+C2sin(βx))y = e^{\alpha x}(C_1 \cos (\beta x) + C_2 \sin (\beta x))

事实上, 直接考虑复值通解:

y=C1e(α+βi)x+C2e(αβi)xy = C_1e^{(\alpha + \beta i)x} + C_2e^{(\alpha - \beta i)x}

C1=C2C_1 = \overline{C_2}时即可得实值通解.


高阶线性齐次ODE

y(n)+i=0n1aiy(i)=f(x)y^{(n)} + \sum_{i=0}^{n-1} a_iy^{(i)} = f(x)

类似于二阶求其特征多项式P(λ)=λn+i=0n1aiλiP(\lambda) = \lambda^n + \sum_{i=0}^{n-1} a_i\lambda^i. 设其有λ1,,λk\lambda_1,\cdots,\lambda_kkk个特征根, 其中λi\lambda_i重数为mim_i, 则其复值通解为:

y(x)=j=1kt=0mi1Cj,txteλjxy(x) = \sum_{j=1}^k \sum_{t=0}^{m_i-1} C_{j,t} \cdot x^t e^{\lambda_j x}


特殊二阶线性非齐次ODE

y+py+qy=Pn(x)eμxy'' + py' + qy = P_n(x)e^{\mu x}

考虑μ\mu的重数mm, 假设一个特解z0=Qn(x)xmeμxz_0 = Q_n(x)x^me^{\mu x}. 解出Qn(x)Q_n(x)后用特解加上其次ODE通解得到非齐次ODE通解.


因此, 任意y+py+qy=Pn(x)f(x)y'' + py' + qy = P_n(x)f(x), 其中f(x)=eaxf(x) = e^{ax}eaxcos(bx)=Re(e(a+bi)x)e^{ax}\cos(bx) = \text{Re}(e^{(a+bi)x})eaxsin(bx)=Im(e(a+bi)x)e^{ax}\sin(bx) = \text{Im}(e^{(a+bi)x})都可以解出.


一般二阶线性非齐次ODE

y+py+qy=f(x)y'' + py' + qy = f(x)

首先求出二阶线性齐次ODE的两个线性无关解y1,y2y_1,y_2, 那么一个特解为:

z0(x)=x0xy1(t)y2(x)y1(x)y2(t)W(y1,y2)(t)f(t)  dtz_0(x) = \int_{x_0}^x \frac{y_1(t)y_2(x) - y_1(x)y_2(t)}{W(y_1,y_2)(t)}f(t) \; \text{d}t

此公式一般很难积分计算, 因此通常计算上一部分的特殊二阶线性非齐次ODE.


欧拉方程

xny(n)+i=0n1aixiy(i)=0x^ny^{(n)} + \sum_{i=0}^{n-1} a_ix^iy^{(i)} = 0

t=lnxt = \ln \vert x \vert, 则:

dydx=1xdydtd2ydx2=ddt(dydx)1dxdt=(1x2xdydt+1xd2ydt2)1x=1x2(d2ydt2dydt)d3ydx3=ddt(d2ydx2)1dxdt=(2x3x(d2ydt2dydt)+1x2(d3ydt3d2ydt2))1x=1x3(d3ydt33d2ydt2+2dydt)\begin{aligned} \frac{\text{d}y}{\text{d}x} &= \frac{1}{x} \cdot \frac{\text{d}y}{\text{d}t} \\ \frac{\text{d}^2y}{\text{d}x^2} &= \frac{\text{d}}{\text{d}t}\left(\frac{\text{d}y}{\text{d}x}\right) \cdot \frac{1}{\frac{\text{d}x}{\text{d}t}} \\ &= \left(-\frac{1}{x^2} \cdot x \cdot \frac{\text{d}y}{\text{d}t} + \frac{1}{x} \cdot \frac{\text{d}^2y}{\text{d}t^2}\right) \cdot \frac{1}{x} \\ &= \frac{1}{x^2} \left( \frac{\text{d}^2y}{\text{d}t^2} - \frac{\text{d}y}{\text{d}t} \right) \\ \frac{\text{d}^3y}{\text{d}x^3} &= \frac{\text{d}}{\text{d}t}\left(\frac{\text{d}^2y}{\text{d}x^2}\right) \cdot \frac{1}{\frac{\text{d}x}{\text{d}t}} \\ &= \left( -\frac{2}{x^3} \cdot x \cdot \left( \frac{\text{d}^2y}{\text{d}t^2} - \frac{\text{d}y}{\text{d}t} \right) + \frac{1}{x^2} \cdot \left( \frac{\text{d}^3y}{\text{d}t^3} - \frac{\text{d}^2y}{\text{d}t^2} \right) \right) \cdot \frac{1}{x} \\ &= \frac{1}{x^3}\left( \frac{\text{d}^3y}{\text{d}t^3} - 3 \frac{\text{d}^2y}{\text{d}t^2}+ 2 \frac{\text{d}y}{\text{d}t} \right) \\ \end{aligned}


一阶线性ODEg

dyidx=fi(x)+j=1naij(x)yj(x)(i{1,2,,n})yj(x0)=ξj(j{1,2,,n})\begin{matrix} \frac{\text{d}y_i}{\text{d}x} = f_i(x) + \sum_{j=1}^n a_{ij}(x)y_j(x) & (i \in \lbrace 1,2, \cdots, n \rbrace) \\ y_j(x_0) = \xi_j & (j \in \lbrace 1,2, \cdots, n \rbrace) \\ \end{matrix}

用向量形式可以表示为:

dYdx=A(x)Y+F(x)\frac{\text{d}\mathbf Y}{\text{d} x} = \mathbf A(x) \mathbf Y + \mathbf F(x)

如果该齐次方程存在nn个线性无关的解, 奇解矩阵为Φ=(Y1,Y2,Y3,,Yn)\mathbf \Phi = (\mathbf Y_1, \mathbf Y_2, \mathbf Y_3, \cdots, \mathbf Y_n), 则通解为ΦC\mathbf{\Phi C}, 其中C=(C1,C2,C3,Cn)\mathbf C = (C_1,C_2,C_3, \cdots C_n)^\top. 一个特解为Z(x)=Φ(x)x0x(Φ(t))1F(t)  dt\mathbf Z(x) =\mathbf \Phi(x) \int_{x_0}^x (\mathbf \Phi (t))^{-1} \mathbf F(t) \;\text{d}t. 因此, 原非齐次方程的通解为:

Y(x)=Φ(x)C+Φ(x)x0x(Φ(t))1F(t)  dt\mathbf Y(x) = \mathbf \Phi(x) \mathbf C + \mathbf \Phi(x) \int_{x_0}^x (\mathbf \Phi (t))^{-1} \mathbf F(t) \;\text{d}t

如何求Φ\mathbf \Phi? 类比一阶线性齐次ODE, 考虑形如Y=eλxr\mathbf Y = e^{\lambda x}\mathbf r的形式: 若A\mathbf Akk个不同的特征值, 特征值λi\lambda_i重数为mim_i, 则对于每个λi\lambda_i存在mim_i个线性无关的解eλixPi(x)e^{\lambda_ix}\mathbf P_i(x), 其中Pi\mathbf P_i为系数为向量的多项式.


向量方程的Wronsky行列式W(x)=det(Φ(x))W(x) = \det(\mathbf \Phi(x)). nn个解线性相关当且仅当W(x)0W(x) \equiv 0. 进一步地, 可以观察到(对行列式求到即对每一行分别求导后求行列式相加):

W(x)=(det(Φ))=an1(x)det(Φ)=an1W(x)W'(x) = (\det(\mathbf \Phi)) = -a_{n-1}(x)\det(\mathbf \Phi) = -a_{n-1}W(x)

W(x)=W(x0)ex0xan1(t)  dtW(x) = W(x_0)e^{-\int_{x_0}^xa_{n-1}(t)\;\text{d}t}.


事实上, 如果令yi=y(i)y_i = y^{(i)}, 一个nn阶线性ODE可以表示为一个一阶nn元线性ODEg:

dyidx=yi+1(i{0,2,,n2})dyn1dx=f(x)iai(x)yiyj(x0)=ξj(j{0,2,,n1})\begin{matrix} \frac{\text{d}y_i}{\text{d}x} = y_{i+1} & (i \in \lbrace 0,2, \cdots, n-2 \rbrace) \\ \frac{\text{d}y_{n-1}}{\text{d}x} = f(x) - \sum_i a_i(x)y_i \\ y_j(x_0) = \xi_j & (j \in \lbrace 0,2, \cdots, n-1 \rbrace) \\ \end{matrix}


求解一阶线性ODEg的常用方法事实上是转为ODE.


技巧

ln\ln求导法

(lnf(x))=f(x)f(x)        (f(x)0)(\ln \vert f(x) \vert)' = \frac{f'(x)}{f(x)}\;\;\;\;(f(x) \ne 0) \\

f(x)=f(x)(lnf(x))f'(x) = f(x) \cdot (\ln \vert f(x) \vert)' \\

分部积分
  1. 对数函数: lnx\ln x, …

  2. 反三角函数: arcsinx\text{arcsin} x, arctanx\text{arctan} x, …

  3. 幂函数: xnx^n, P(x)P(x), …

  4. 三角函数: sinx\sin x, cosx\cos x, …

  5. 指数函数: exe^x, …

上述函数结合时, 排名靠前的适合求导, 留在原地准备分部积分后求导; 排名靠后的适合积分, 积分放入d\text{d}后.


换元

ax2+bx+c=±ax+t\sqrt{ax^2+bx+c} = \pm \sqrt{a}x + t

ax2+bx+c=tx±c\sqrt{ax^2+bx+c} = tx \pm \sqrt{c}

a(xb)(xc)=t(xb)\sqrt{a(x-b)(x-c)} = t(x-b)

ax+bcx+dn=t\sqrt[n]{\frac{ax+b}{cx+d}} = t

被积函数关于sinx\sin x奇函数: cosx=t\cos x = t.

被积函数关于cosx\cos x奇函数: sinx=t\sin x = t.

被积函数关于sinx\sin x, cosx\cos x都为偶函数: tanx=t\tan x = t.


x2+a2\sqrt{x^2+a^2}: x=ashtx = a \text{sh} tx=atantx = a \tan t.

x2a2\sqrt{x^2-a^2}: x=achtx = a \text{ch} tx=±asectx = \pm a \sec t.

a2x2\sqrt{a^2-x^2}: x=asintx = a \sin t.


三角有理函数积分

asinx+bcosxasinx+bcosx  dx=dx=x+C\int \frac{a \sin x + b \cos x}{a \sin x + b \cos x} \;\text{d}x = \int \text{d}x = x + C \\

bsinx+acosxasinx+bcosx  dx=lnasinx+bcosx+C\int \frac{-b \sin x + a \cos x}{a \sin x + b \cos x} \;\text{d}x = \ln \vert a \sin x + b \cos x \vert + C \\

asinx+bcosxa\sin x + b \cos xbsinx+acosx-b \sin x + a \cos x线性无关, 因此可以求出所有形如下方的积分:

csinx+dcosxasinx+bcosx  dx\int \frac{c \sin x + d \cos x}{a \sin x + b \cos x} \;\text{d}x


鸣谢

感谢何昊天学长的微积分复习讲座(手动@微信公众号:乐学)。


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文章作者: Magolor
文章链接: https://magolor.cn/2019/12/28/2019-12-28-blog-01/
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