人工智能教程 - 数学基础课程1.1 - 数学分析（一）15-17 微分方程和分离变量,定积分及性质,微积分第一定理
微分方程 differential equation
Ex:
dydx=f(x)\frac{dy}{dx} = f(x)dxdy=f(x)
y=∫f(x)dxy= \int f(x) dxy=∫f(x)dx
solved substitution
Ex2:
(ddx+x)(\frac{d}{dx}+x)(dxd+x)为annihilation operator 湮没算符 in quantum mechnics
dydx=−xy\frac{dy}{dx} = -xydxdy=−xy
dyy=−xdx\frac{dy}{y} = -xdxydy=−xdx
∫dyy=−∫xdx\int \frac{dy}{y} = -\int xdx∫ydy=−∫xdx
lny=−x2/2+Cln y = -x^2/2 +Clny=−x2/2+C
elny=e−x2/2+Ce^{lny} = e^{-x^2/2}+Celny=e−x2/2+C
y=Ae−x2/2(A=ec)y = Ae^{-x^2/2} (A=e^c)y=Ae−x2/2(A=ec)
Solution:
y=ae−x2/2 any ay = ae^{-x^2/2} \ any \ ay=ae−x2/2 any a
dydx=a.ddx.e−x2/2\frac{dy}{dx} = a.\frac{d}{dx} . e^{-x^2/2}dxdy=a.dxd.e−x2/2
=a.(−x).e−x2/2= a.(-x) . e^{-x^2/2}=a.(−x).e−x2/2
=−x.y= -x.y=−x.y
by the way,the function is known as the normal distribution 正态分布
分离变量法
Only works if u’ does not change sign
作者：KuFun人工智能
SEPARATION OF VARIABLES
dydx=f(x).g(y)\frac{dy}{dx} = f(x).g(y)dxdy=f(x).g(y) dyg(y)=f(x)dx\frac{dy}{g(y)} = f(x)dxg(y)dy=f(x)dx H(y)=∫dyg(y);F(x)=∫f(x)dxH(y) = \int \frac{dy}{g(y)}; F(x) = \int f(x)dxH(y)=∫g(y)dy;F(x)=∫f(x)dx H(y)=F(x)+C→implicitH(y)=F(x)+C \rightarrow implicitH(y)=F(x)+C→implicit y=H−1(F(x)+C)y = H^{-1}(F(x)+C)y=H−1(F(x)+C) 定积分Definite Integrals
Find Area under a curve =∫abf(x)dx\int_{a}^{b} f(x) dx∫abf(x)dx累积和(cumulative sum)
To compute the area divide into rectangles add up areas take the limit as rectangles get thin 简写 abbreviation∑i=1nai=a1+a2+...+an\sum_{i=1}^{n} a_i = a_1+a_2+...+a_n∑i=1nai=a1+a2+...+an
∑\sum∑ is called sigma
Notation (Riemann Sums)General Procedure for definite integrals:
Δx=b−an\Delta x = \frac{b-a}{n}Δx=nb−a Pick any height of f in each interval: ∑i=1nf(ci)Δx→∫abf(x)dx{\color{Red} \sum_{i=1}^{n} f(c_i) \Delta x\rightarrow \int_{a}^{b} f(x)dx}∑i=1nf(ci)Δx→∫abf(x)dx 微积分第一定理 Fundamental theorem of calculus(FTC1) If F’(x) = f(x), then ∫abf(x)dx=F(b)−F(a)=F(x)∣ab{\color{Red} {\color{Red} \int_{a}^{b} f(x)dx = F(b)-F(a)}=F(x)|_{a}^{b}}∫abf(x)dx=F(b)−F(a)=F(x)∣ab F=∫f(x)dxF=\int f(x)dxF=∫f(x)dx NOTATION F(b)−F(a)=F(x)∣ab=F(x)∣x=ax=bF(b) - F(a) = F(x)|_{a}^{b} = F(x)|_{x=a}^{x=b}F(b)−F(a)=F(x)∣ab=F(x)∣x=ax=b Ex1: F(x)=x3/3F(x) = x^3/3F(x)=x3/3 F′(x)=x2(=f(x))F'(x) = x^2(=f(x))F′(x)=x2(=f(x)) ∫abx2dx=F(b)−F(a)=b33−a33\int_{a}^{b}x^2dx = F(b)-F(a)=\frac{b^3}{3}-\frac{a^3}{3}∫abx2dx=F(b)−F(a)=3b3−3a3 ∫0bx2dx=x33∣0b=b33\int_{0}^{b}x^2dx = \frac{x^3}{3}|_{0}^{b}=\frac{b^3}{3}∫0bx2dx=3x3∣0b=3b3 True geometric interp of definit integral is that area above x-axis minus the area below the x-axis 定积分的性质Properties of integrals
1.∫ab(f(x)+g(x))dx=∫abf(x)dx+∫abg(x)dx1.\int_{a}^{b}(f(x)+g(x))dx = \int_{a}^{b}f(x)dx +\int_{a}^{b}g(x)dx1.∫ab(f(x)+g(x))dx=∫abf(x)dx+∫abg(x)dx 2.∫abCf(x)dx=C∫abf(x)dx2.\int_{a}^{b}Cf(x)dx = C\int_{a}^{b}f(x)dx2.∫abCf(x)dx=C∫abf(x)dx 3.∫abf(x)dx+∫bcf(x)dx=∫acf(x)dx (a<b<c)3.\int_{a}^{b}f(x)dx +\int_{b}^{c}f(x)dx = \int_{a}^{c}f(x)dx \ \ (a<b<c)3.∫abf(x)dx+∫bcf(x)dx=∫acf(x)dx (a<b<c) 4.∫aaf(x)dx=04.\int_{a}^{a}f(x)dx = 04.∫aaf(x)dx=0 5.∫abf(x)dx=−∫baf(x)dx5.\int_{a}^{b}f(x)dx = -\int_{b}^{a}f(x)dx5.∫abf(x)dx=−∫baf(x)dx 6. (Estimation)If f(x)<=g(x),then:∫abf(x)dx≤∫abg(x)dx\int_{a}^{b}f(x)dx \leq \int_{a}^{b}g(x)dx∫abf(x)dx≤∫abg(x)dx change of variables: (=substituion) ∫u1u2g(u)dx=∫x1x2g(u(x)).u′(x)dx{\color{Red} \int_{u_1}^{u_2}g(u)dx = \int_{x_1}^{x_2}g(u(x)).u'(x)dx}∫u1u2g(u)dx=∫x1x2g(u(x)).u′(x)dx| u = u(x) | u1 = u(x1) |
|---|---|
| du = u’(x)dx | u2 = u(x2) |
作者：KuFun人工智能
