From 6fb06fbb2be445a07b3c9fc13c077a1de9b91f78 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 19 Jul 2021 23:29:30 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E6=97=A0=E7=A9=B7=E7=BA=A7?= =?UTF-8?q?=E6=95=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../8-infinite-series/infinite-series.pdf | Bin 0 -> 12511 bytes .../8-infinite-series/infinite-series.tex | 41 ++++ .../differential-equation.pdf | Bin 0 -> 86912 bytes .../differential-equation.tex | 195 ++++++++++++++++++ .../8-infinite-series/infinite-series.pdf | Bin 12511 -> 83908 bytes .../8-infinite-series/infinite-series.tex | 158 +++++++++++++- .../differential-equation.pdf | Bin 103928 -> 120947 bytes .../differential-equation.tex | 88 +++++++- 8 files changed, 471 insertions(+), 11 deletions(-) create mode 100644 advanced-math/exercise/8-infinite-series/infinite-series.pdf create mode 100644 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UZW2i#do;=x12Qpj@j3ze17!(>ssI20 literal 0 HcmV?d00001 diff --git a/advanced-math/exercise/9-differential-equation/differential-equation.tex b/advanced-math/exercise/9-differential-equation/differential-equation.tex new file mode 100644 index 0000000..abca49b --- /dev/null +++ b/advanced-math/exercise/9-differential-equation/differential-equation.tex @@ -0,0 +1,195 @@ +\documentclass[UTF8, 12pt]{ctexart} +% UTF8编码,ctexart现实中文 +\usepackage{color} +% 使用颜色 +\definecolor{orange}{RGB}{255,127,0} +\definecolor{violet}{RGB}{192,0,255} +\definecolor{aqua}{RGB}{0,255,255} +\usepackage{geometry} +\setcounter{tocdepth}{4} +\setcounter{secnumdepth}{4} +% 设置四级目录与标题 +\geometry{papersize={21cm,29.7cm}} +% 默认大小为A4 +\geometry{left=3.18cm,right=3.18cm,top=2.54cm,bottom=2.54cm} +% 默认页边距为1英尺与1.25英尺 +\usepackage{indentfirst} +\setlength{\parindent}{2.45em} +% 首行缩进2个中文字符 +\usepackage{setspace} +\renewcommand{\baselinestretch}{1.5} +% 1.5倍行距 +\usepackage{amssymb} +% 因为所以 +\usepackage{amsmath} +% 数学公式 +\usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} +% 超链接 +\author{Didnelpsun} +\title{微分方程} +\date{} +\begin{document} +\maketitle +\pagestyle{empty} +\thispagestyle{empty} +\tableofcontents +\thispagestyle{empty} +\newpage +\pagestyle{plain} +\setcounter{page}{1} + +\section{一阶微分方程} + +\subsection{可分离变量微分方程} + +\textbf{例题:}求$y\sin\dfrac{x}{2}\,\textrm{d}x-\cos\dfrac{x}{2}\,\textrm{d}y=0$的通解。 + +解:$\dfrac{\textrm{d}y}{\textrm{d}x}=y\tan\dfrac{x}{2}$,$\dfrac{\textrm{d}y}{y}=\tan\dfrac{x}{2}\,\textrm{d}x$,$\displaystyle{\int\dfrac{\textrm{d}y}{y}=2\int\tan\dfrac{x}{2}\,\textrm{d}\dfrac{x}{2}}$。 + +解得$\ln\vert y\vert=-\ln\left(\cos\dfrac{x}{2}\right)^2+\ln C_1$(取对数更好解),$\vert y\vert=\dfrac{C_1}{\left(\cos\dfrac{x}{2}\right)^2}$。 + +$y=\dfrac{\pm C_1}{\left(\cos\dfrac{x}{2}\right)^2}$,令$C=\pm C_1$,得$y=\dfrac{C}{1+\cos x}$。 + +注意在第一步时将$y$除到分母上,本来$y$为任意常数,变为$y\neq0$,所以解得最后$C\neq0$,而实际上$y$可以为0,所以$C$应该为任意常数。 + +此时解为全部解,为通解加上$y=0$的奇解。 + +\subsection{多项式换元} + +\textbf{例题:}求微分方程$\textrm{d}y=\sin(x+y+100)\,\textrm{d}x$的通解。 + +解:令$u=x+y+100$,$\dfrac{\textrm{d}u}{\textrm{d}x}=1+\dfrac{\textrm{d}y}{\textrm{d}x}$,$\dfrac{\textrm{d}y}{\textrm{d}x}=\sin(x+y+100)$,$\therefore\dfrac{\textrm{d}u}{\textrm{d}x}=1+\sin u$。 + +$\dfrac{\textrm{d}u}{1+\sin u}=\textrm{d}x$,$\displaystyle{\int\dfrac{\textrm{d}u}{1+\sin u}}=\int\textrm{d}x$,$\displaystyle{\int\dfrac{1-\sin u}{\cos^2u}}\textrm{d}u=x$。 + +$\int\sec^2u-\tan u\sec u\,\textrm{d}u=x$,即$\tan u-\sec u=x+C$。代回$u=x+y+100$: + +通解$\tan(x+y+100)-\sec(x+y+100)=x+C$。 + +所有解:$\tan(x+y+100)-\sec(x+y+100)=x+C$,$x+y+100=2k\pi-\dfrac{\pi}{2}$。 + +\subsection{自然齐次方程} + +\textbf{例题:}设$L$是一条平面曲线,其上任意一点$P(x,y)$($x>0$)到坐标原点的距离恒等于该点处的切线在$y$轴上的截距,且$L$经过点$\left(\dfrac{1}{2},0\right)$,求$L$的方程。 + +解:$(x,y)$到坐标原点的距离为$\sqrt{x^2+y^2}$。 + +若$y=y(x)$,则切线为$Y-y=y'(X-x)$,令$X=0$,解得$Y=y-xy'$。 + +$\therefore\sqrt{x^2+y^2}=y-xy'$,解得$y'=\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{y-\sqrt{x^2+y^2}}{x}=\dfrac{y}{x}-\sqrt{1+\dfrac{y^2}{x^2}}$。 + +令$\dfrac{y}{x}=u$,则$y=ux$,$\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{\textrm{d}u}{\textrm{d}x}x+u$。代入$y'$: + +$\dfrac{\textrm{d}u}{\textrm{d}x}x+u=u-\sqrt{1+u^2}$,$\dfrac{\textrm{d}u}{\sqrt{1+u^2}}=-\dfrac{\textrm{d}x}{x}$,$\displaystyle{\int\dfrac{\textrm{d}u}{\sqrt{1+u^2}}=-\int\dfrac{\textrm{d}x}{x}}$。 + +$\therefore\ln(u+\sqrt{1+u^2})=-\ln x+\ln C$,$u+\sqrt{1+u^2}=\dfrac{C}{x}$。 + +代入$\dfrac{y}{x}+\sqrt{1+\dfrac{y^2}{x^2}}=\dfrac{C}{x}$,$y+\sqrt{x^2+y^2}=C$。 + +\subsection{一阶线性方程} + +\textbf{例题:}求微分方程$y'+1=e^{-y}\sin x$的通解。 + +解:已知对$e^{-y}\sin x$无法处理,所以必然需要对其转换,$e^yy'+e^y=\sin x$。 + +$\therefore(e^y)'+e^y=\sin x$,令$e^y=u$,$u'+u=\sin x$,$P(x)=1$,$Q(x)=\sin x$。 + +$e^y=u=e^{-\int\textrm{d}x}(\int e^{\int\textrm{d}x}\sin x\,\textrm{d}x+C)=e^{-x}(\int e^x\sin x\,\textrm{d}x+C)$,积分再现表格解出$\int e^x\sin x\,\textrm{d}x$:$=e^{-x}\left(\dfrac{1}{2}e^x(\sin x-\cos x)+C\right)$。 + +\subsection{伯努利方程} + +\textbf{例题:}求$y\,\textrm{d}x=(1+x\ln y)x\,\textrm{d}y$($y>0$)的通解。 + +解:将导数放到一边:$\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{y}{(1+x\ln y)x}$,这个算式无法处理。 + +而颠倒$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{(1+x\ln y)x}{y}=\dfrac{1}{y}x+\dfrac{\ln y}{y}x^2$。 + +凑伯努利方程:$x'+P(x)x=Q(x)x^n$:$x'-\dfrac{1}{y}x=\dfrac{\ln y}{y}x^2$。$P(x)=-\dfrac{1}{y}$,$Q(x)=\dfrac{\ln y}{y}$。 + +乘$x^{-2}$降阶:$x^{-2}x'-\dfrac{1}{y}x^{-1}=\dfrac{\ln y}{y}$。令$z=x^{-1}$,$\dfrac{\textrm{d}z}{\textrm{d}y}=-\dfrac{1}{x^2}\dfrac{\textrm{d}x}{\textrm{d}y}$。代入方程: + +$-\dfrac{\textrm{d}z}{\textrm{d}y}-\dfrac{1}{y}z=\dfrac{\ln y}{y}$,$\dfrac{\textrm{d}z}{\textrm{d}y}+\dfrac{1}{y}z=-\dfrac{\ln y}{y}$,利用公式: + +$z=e^{-\int\frac{1}{y}\textrm{d}y}\left(\displaystyle{\int e^{\int\frac{1}{y}\textrm{d}y}\cdot\left(\dfrac{\ln y}{y}\right)+C}\right)=\dfrac{1}{y}(-\int\ln y\,\textrm{d}y+C)=\dfrac{1}{y}(-y(\ln y-1)+C)=-\ln y+1+\dfrac{C}{y}$。 + +$\therefore x=\dfrac{y}{-y\ln y+y+C}$。 + +\section{二阶可降阶微分方程} + +\subsection{\texorpdfstring{$y''=f(x,y')$}\ 型} + +\textbf{例题:}求$y''=\dfrac{2xy'}{1+x^2}$的通解。 + +解:令$y'=p$,$p'=\dfrac{2xp}{1+x^2}$,$\dfrac{\textrm{d}p}{\textrm{d}x}=\dfrac{2xp}{1+x^2}$,$\dfrac{\textrm{d}p}{p}=\dfrac{2x}{1+x^2}$,$\displaystyle{\int\dfrac{\textrm{d}p}{p}=\int\dfrac{2x}{1+x^2}}$。 + +$\ln\vert p\vert=\ln(1+x^2)+\ln C_1$,$p=\pm C_1(1+x^2)=C_2(1+x^2)$。 + +$y'=C(1+x^2)$,$\therefore y=C_2\left(x+\dfrac{x^3}{3}+x\right)+C$。 + +\subsection{\texorpdfstring{$y''=f(y,y')$}\ 型} + +\section{高阶线性微分方程} + +\subsection{常系数齐次线性微分方程} + + + +\subsection{常系数非齐次线性微分方程} + +先将常系数非齐次线性微分方程变为常系数齐次线性微分方程求解,然后加上非齐次方程的一个特解,就是非齐次方程的一个通解。 + +\textbf{例题:}求$y''-4y'+4y=3xe^{2x}$的通解。 + +解:变为常系数齐次线性微分方程:$y''-4y'+4y$。 + +写出特征方程:$\lambda^2-4\lambda+4=0$,从而$(\lambda-2)^2=0$,$\lambda_1=\lambda_2=2$。 + +从而$y$齐次方程的通解为$(C_1+C_2x)e^{2x}$。 + +根据特解的设置方法,设$y^*=e^{2x}(ax+b)x^2$。 + +代回二阶方程,$a=\dfrac{1}{2}$,$b=0$。通解为$(C_1+C_2x)e^{2x}+\dfrac{1}{2}x^3e^{2x}$。 + +\textbf{例题:}微分方程$y''-4y'+3y=e^x\cos x+xe^{3x}$的通解。 + +解:首先常系数齐次线性微分方程:$y''-4y'+3y=0$。 + +特征方程为$\lambda^2-4\lambda+3=0$,解得特征值为$\lambda_1=1$,$\lambda_2=3$。 + +所以该齐次方程的通解:$y=C_1e^x+C_2e^{3x}$。 + +然后求特解,首先求后面$f_2(x)=xe^{3x}$的特解$y_2^*$。 + +根据公式因为$\alpha$为单特征根,即$\aleph=3=\lambda_2\neq\lambda_1$,所以$y_2^*=e^{3x}(ax+b)x$。 + +然后是求$f_1(x)=e^x\cos x$的特解$y_1^*$。 + +其中$P_m(x)=1$,$P_n(x)=0$,$l=0$。所以设$P_m(x)=A$,$P_n(x)=B$。 + +对$k$,自由项中$\alpha=\beta=1$,得到$1\pm i$。又$1\pm i\neq\lambda_1=1\neq\lambda_2=3$,$k=0$。 + +最后$y_1^*=e^x(A\cos x+B\sin x)$。通解为$y=C_1e^x+C_2e^{3x}+e^x(A\cos x+B\sin x)+e^{3x}(ax+b)x$。 + +\section{微分方程概念} + +对于有些方程并不需要求解后才能解决问题。 + +\subsection{已知微分方程的解反求系数} + +\textbf{例题:}设$y_1,y_2$为一阶非齐次线性微分方程$y'+p(x)y=q(x)$的两个特解,若常数$\lambda,\mu$使得$\lambda y_1+\mu y_2$是该方程的解,$\lambda y_1-\mu y_2$是该方程对应的齐次方程的解,则()。 + +$A.\lambda=\dfrac{1}{2},\mu=\dfrac{1}{2}$\qquad$B.\lambda=-\dfrac{1}{2},\mu=-\dfrac{1}{2}$\qquad$C.\dfrac{2}{3},\mu=\dfrac{1}{3}$\qquad$\lambda=\dfrac{2}{3},\mu=\dfrac{2}{3}$ + +\subsection{不解微分方程,利用方程隐含信息} + +$F(y,y',y'',\cdots,y^{(n)})=0$反映了\textbf{未知函数及其各阶导数之间的关系}。 + +\textbf{例题:}设$y=f(x)$是方程$y''-2y'+4y=0$的一个解,若$f(x_0)>0$,且$f'(x_0)=0$,则函数$f(x)$在点$x_0$()。 + +$A.$取得最大值\qquad$B.$取得最小值\qquad$C.$某个邻域内单调增加\qquad$D.$某个邻域内单调减少 + +解:因为$y=f(x)$是方程$y''-2y'+4y=0$的一个解,所以直接代入$x_0$:$y''(x_0)-2y'(x_0)+4y(x_0)=0$。又$f'(x_0)=0$。 + +$y''(x_0)=-4y(x_0)<0$,所以该点为极大值点。 + +\end{document} diff --git a/advanced-math/knowledge/8-infinite-series/infinite-series.pdf b/advanced-math/knowledge/8-infinite-series/infinite-series.pdf index a0062c3c088b4feb690816b3fe388d2cdf6020b0..e944727b327af266d637cfab7cfb299de36df1ac 100644 GIT binary patch literal 83908 zcma&NQ;=p|7OkDO?MmDErfs{@wr$(CZB^Q~U1{64Pyc_Mi|*)-6W`5>eY@uxd&V>N z9Ft5zM2wD!o)w1d@b>BzhJ}@okCdl 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x>0$,$x>\ln(1+x)$,$\therefore\dfrac{1}{n}>\ln\left(1+\dfrac{1}{n}\right)$。 + +又对于$\ln\left(1+\dfrac{1}{n}\right)=\ln\dfrac{n+1}{n}=\ln(n+1)-\ln n$。 + +$S_n=\ln\dfrac{2}{1}+\ln\dfrac{3}{2}+\cdots+\ln\dfrac{n+1}{n}=\ln2-\ln1+\ln3-\ln2+\cdots+\ln(n+1)-\ln n=\ln(n+1)$。当$\ln(n+1)$在$n\to\infty$时,$S_n\to\infty$。 + +所以$\sum\limits_{n=1}^\infty\ln\left(1+\dfrac{1}{n}\right)$发散,则$\sum\limits_{n=1}^\infty\dfrac{1}{n}$也发散。 + +\textcolor{aqua}{\textbf{定理:}}$p$级数:$\sum\limits_{n=1}^\infty\dfrac{1}{n^p}\left\{\begin{array}{l} + p>1, \text{收敛} \\ + p\leqslant1, \text{发散} +\end{array}\right.$。 + +\paragraph{比较判别法极限性质} \leavevmode \medskip + +是比较判别法的推论,利用极限的阶数来比较。 + +给出两个正项级数$\sum\limits_{n=1}^\infty u_n$,$\sum\limits_{n=1}^\infty v_n$,$v_n\neq0$,且$\lim\limits_{n\to\infty}\dfrac{u_n}{v_n}=A$: + +\begin{enumerate} + \item 若$A=0$,则当$\sum\limits_{n=1}^\infty v_n$收敛时,$\sum\limits_{n=1}^\infty u_n$也收敛。 + \item 当$A=+\infty$,当$\sum\limits_{n=1}^\infty v_n$发散时,$\sum\limits_{n=1}^\infty u_n$也发散。 + \item 若$01$,所以$\sum\limits_{n=1}^\infty\left(\dfrac{1}{n}-\sin\dfrac{1}{n}\right)$收敛。 + +\paragraph{比值判别法} \leavevmode \medskip + +也称为达朗贝尔判别法。 + +\textcolor{aqua}{\textbf{定理:}}给出一正项级数$\sum\limits_{n=1}^\infty u_n$,若$\lim\limits_{n\to\infty}\dfrac{u_{n+1}}{u_n}=\rho$,则:\ding{172}若$\rho<1$,则$\sum\limits_{n=1}^\infty u_n$收敛;\ding{173}若$\rho>1$,则$\sum\limits_{n=1}^\infty u_n$发散。 + +\textcolor{orange}{注意:}$\rho=1$时无法根据此判断$\sum\limits_{n=1}^\infty u_n$敛散性,如$\sum\limits_{n=1}^\infty\dfrac{1}{n}$发散,但$\sum\limits_{n=1}^\infty\dfrac{1}{n^2}$收敛。 + +适用于含有$a^n$,$n!$,$n^n$的通项。 + +\textbf{例题:}判断级数$\sum\limits_{n=1}^\infty\dfrac{\vert a\vert^nn!}{n^n}$的敛散性,其中$a$为非零常数。 + +解: + +\paragraph{根值判别法} \leavevmode \medskip + +也称为柯西判别法。 + +\textcolor{aqua}{\textbf{定理:}}$\sum\limits_{n=1}^\infty u_n$ + +\subsubsection{交错级数} + +\paragraph{概念} \leavevmode \medskip + +\paragraph{莱布尼兹判别法} \leavevmode \medskip + +\subsubsection{任意项级数} + +\paragraph{绝对收敛} \leavevmode \medskip + +\paragraph{条件收敛} \leavevmode \medskip + +\section{幂级数} + +\subsection{概念与收敛域} + +\subsubsection{概念} + +\subsubsection{阿贝尔定理} + +\subsubsection{收敛域} + +\subsection{幂级数求和函数} + +\subsubsection{概念} + +\subsubsection{运算法则} + +\subsubsection{性质} + +\subsubsection{重要展开式} + +\subsection{函数展开为幂级数} + +\subsubsection{概念} + +\subsubsection{求法} + +\paragraph{直接法} \leavevmode \medskip + +\paragraph{间接法} \leavevmode \medskip \end{document} diff --git a/advanced-math/knowledge/9-differential-equation/differential-equation.pdf b/advanced-math/knowledge/9-differential-equation/differential-equation.pdf index 2df274c9a4565774f7642e3733feaa85b9d346fc..2f91a87cf71a6faa078eb514f5f7a84f893ca416 100644 GIT binary patch delta 116668 zcmZs>V{;}9)U6rYwr$(i9iwC0cJA2h*h$B>ZQD*d?$}PwbEamh=AAiTcI_XqYt_2e z+OJhm5B1Qfl*$rPjI2zYaFj>4SFdpFJV~e&KvouRb|E1+S2q`PV|zHS%`2U$_$@As zofrD%W^9X_WaQ?VKO*=AU0_YNorw%pRaj?CZyyHU32<=woO?gfh(Ql|&E9x;v43$oHXuDX zfB`E%cjPv}z;U83ag+V)oqVOfOhRre(~%{s)7-WBO9?$QJ{{0`P?7Q45Pe^eR^KQP 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若$p^2-4q=0$,设$\lambda_1,\lambda_2$是特征方程的两个相等实根,即二重根,令$\lambda=\lambda_1=\lambda_2$,其通解为$y=(C_1+C_2x)e^{\lambda x}$。 + \item 若$p^2-4q<0$,设$\alpha\pm\beta i$是特征方程的一对共轭复根,$\lambda_{1,2}=\dfrac{-p\pm\sqrt{4q-p^2}i}{2}$\\$=-\dfrac{p}{2}\pm\dfrac{\sqrt{4q-p^2}}{2}i$,记为$\alpha\pm\beta i$,其通解为$y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)$。 +\end{enumerate} + +\subsection{二阶常系数非齐次线性微分方程的特解} + +设$P_n(x)$,$P_m(x)$分别为$x$的$n$次$m$次多项式。 + +\begin{enumerate} + \item 当自由项$f(x)=P_n(x)e^{\alpha x}$时,特解设为$y^*=e^{\alpha x}Q_n(x)x^k$,其中$e^{\alpha x}$照抄,$Q_n(x)$为$x$的$n$次多项式,且$k=\left\{\begin{array}{ll} + 0, & \alpha\text{不是特征根} \\ + 1, & \alpha\text{是单特征根} \\ + 2, & \alpha\text{是二重特征根} + \end{array}\right.$。 + \item 当自由项$f(x)=e^{\alpha x}[P_m(x)\cos\beta x+P_n(x)\sin\beta x]$时,特解设为$y^*=e^{\alpha x}[Q_l^{(1)}(x)\cos\beta x+Q_l^{(2)}(x)\sin\beta x]x^k$,其中$e^{\alpha x}$照抄,$l=\max\{m,n\}$,$Q_l^{(1)}$、$Q_l^{(2)}$为$x$的两个不同的$l$次多项式,且$k=\left\{\begin{array}{ll} + 0, & \alpha\pm\beta i\text{不是特征根} \\ + 1, & \alpha\pm\beta i\text{是特征根} \\ + \end{array}\right.$。 +\end{enumerate} + \end{document}