tex源代码如下:
1 documentclass[a4paper, 12pt]{article} % Font size (can be 10pt, 11pt or 12pt) and paper size (remove a4paper for US letter paper) 2 usepackage{amsmath,amsfonts,bm} 3 usepackage{hyperref} 4 usepackage{amsthm} 5 usepackage{amssymb} 6 usepackage{framed,mdframed} 7 usepackage{graphicx,color} 8 usepackage{mathrsfs,xcolor} 9 usepackage[all]{xy} 10 usepackage{fancybox} 11 usepackage{xeCJK} 12 ewtheorem{adtheorem}{定理} 13 setCJKmainfont[BoldFont=FZYaoTi,ItalicFont=FZYaoTi]{FZYaoTi} 14 definecolor{shadecolor}{rgb}{1.0,0.9,0.9} %背景色为浅红色 15 ewenvironment{theorem} 16 {egin{mdframed}[backgroundcolor=gray!40,rightline=false,leftline=false,topline=false,bottomline=false]egin{adtheorem}} 17 {end{adtheorem}end{mdframed}igskip} 18 ewtheorem*{bdtheorem}{定义} 19 ewenvironment{definition} 20 {egin{mdframed}[backgroundcolor=gray!40,rightline=false,leftline=false,topline=false,bottomline=false]egin{bdtheorem}} 21 {end{bdtheorem}end{mdframed}igskip} 22 ewtheorem*{cdtheorem}{习题} 23 ewenvironment{exercise} 24 {egin{mdframed}[backgroundcolor=gray!40,rightline=false,leftline=false,topline=false,bottomline=false]egin{cdtheorem}} 25 {end{cdtheorem}end{mdframed}igskip} 26 ewtheorem{ddtheorem}{注} 27 ewenvironment{remark} 28 {egin{mdframed}[backgroundcolor=gray!40,rightline=false,leftline=false,topline=false,bottomline=false]egin{ddtheorem}} 29 {end{ddtheorem}end{mdframed}igskip} 30 ewtheorem{edtheorem}{引理} 31 ewenvironment{lemma} 32 {egin{mdframed}[backgroundcolor=gray!40,rightline=false,leftline=false,topline=false,bottomline=false]egin{edtheorem}} 33 {end{edtheorem}end{mdframed}igskip} 34 usepackage[protrusion=true,expansion=true]{microtype} % Better typography 35 usepackage{wrapfig} % Allows in-line images 36 usepackage{mathpazo} % Use the Palatino font 37 usepackage[T1]{fontenc} % Required for accented characters 38 linespread{1.05} % Change line spacing here, Palatino benefits from a slight increase by default 39 40 makeatletter 41 enewcommand@biblabel[1]{ extbf{#1.}} % Change the square brackets for each bibliography item from '[1]' to '1.' 42 enewcommand{@listI}{itemsep=0pt} % Reduce the space between items in the itemize and enumerate environments and the bibliography 43 44 enewcommand{maketitle}{ % Customize the title - do not edit title 45 % and author name here, see the TITLE block 46 % below 47 enewcommand efname{参考文献} 48 ewcommand{D}{displaystyle} ewcommand{ i}{Rightarrow} 49 ewcommand{ds}{displaystyle} enewcommand{ i}{ oindent} 50 ewcommand{pa}{partial} ewcommand{Om}{Omega} 51 ewcommand{om}{omega} ewcommand{sik}{sum_{i=1}^k} 52 ewcommand{vov}{VertomegaVert} ewcommand{Umy}{U_{mu_i,y^i}} 53 ewcommand{lamns}{lambda_n^{^{scriptstylesigma}}} 54 ewcommand{chiomn}{chi_{_{Omega_n}}} 55 ewcommand{ullim}{underline{lim}} ewcommand{sy}{oldsymbol} 56 ewcommand{mvb}{mathversion{bold}} ewcommand{la}{lambda} 57 ewcommand{La}{Lambda} ewcommand{va}{varepsilon} 58 ewcommand{e}{eta} ewcommand{al}{alpha} 59 ewcommand{dis}{displaystyle} ewcommand{R}{{mathbb R}} 60 ewcommand{N}{{mathbb N}} ewcommand{cF}{{mathcal F}} 61 ewcommand{gB}{{mathfrak B}} ewcommand{eps}{epsilon} 62 egin{flushright} % Right align 63 {LARGE@title} % Increase the font size of the title 64 65 vspace{50pt} % Some vertical space between the title and author name 66 67 {large@author} % Author name 68 \@date % Date 69 70 vspace{40pt} % Some vertical space between the author block and abstract 71 end{flushright} 72 } 73 74 % ---------------------------------------------------------------------------------------- 75 % TITLE 76 % ---------------------------------------------------------------------------------------- 77 78 itle{ extbf{《常微分方程教程》习题2-2,4\[2em]一个跟踪问题}} 79 80 author{small{叶卢庆}\{small{杭州师范大学理学院,学号:1002011005}}\{small{Email:h5411167@gmail.com}}} % Institution 81 enewcommand{ oday}{ umberyear. umbermonth. umberday} 82 date{ oday} % Date 83 84 % ---------------------------------------------------------------------------------------- 85 86 egin{document} 87 maketitle % Print the title section 88 89 % ---------------------------------------------------------------------------------------- 90 % ABSTRACT AND KEYWORDS 91 % ---------------------------------------------------------------------------------------- 92 93 % enewcommand{abstractname}{摘要} % Uncomment to change the name of the abstract to something else 94 95 % egin{abstract} 96 97 % end{abstract} 98 99 % hspace*{3,6mm} extit{关键词:} % Keywords 100 101 % vspace{30pt} % Some vertical space between the abstract and first section 102 103 % ---------------------------------------------------------------------------------------- 104 % ESSAY BODY 105 % ---------------------------------------------------------------------------------------- 106 egin{exercise}[2-2,4] 107 跟踪:设某 $A$ 从 $Oxy$ 平面的原点出发,沿 $x$ 轴正方向前进;同时某 $B$ 108 从点 $(0,b)$ 开始跟踪 $A$,即 $B$ 的运动方向永远指向 $A$ 并与 $A$ 保持 109 等距 $b$.试求 $B$ 的光滑运动轨迹. 110 end{exercise} 111 egin{proof}[解] 112 设在时刻 $t$ 的时候 $A$ 位于 $(f(t),0)$.其中 $f(0)=0$,且 $f(t)$ 是关于 113 $t$ 的严格单调增函数.设在时刻 $t$ 的 114 $B$ 位于 $(P(t),Q(t))$,其中 $P(0)=0,Q(0)=b$.不妨设 $b eq 0$,否则 $B$ 115 的运动将与 $A$ 重合,这是没什么意思的,再根据对称性不妨设 $b>0$.且由于 $B$ 的路径光滑,因此关于 116 $t$ 的函数 $P,Q$ 都是连续可微的.由于 $B$ 的方向一直指向 $A$,因此 117 egin{equation} 118 label{eq:10.51} 119 (P'(t),Q'(t))=k(f(t)-P(t),-Q(t)). 120 end{equation} 121 其中 $k>0$.由于 $A,B$ 间距始终为 $b$,因此 122 egin{equation} 123 label{eq:10.52} 124 [P(t)-f(t)]^2+Q(t)^2=b^2. 125 end{equation} 126 当 $Q(t) eq 0$ 时,$Q'(t)$ 也不为0.此时 将(1) 代入 (2) 可得 127 egin{equation} 128 label{eq:11.02} 129 (P'(t))^2+(Q'(t))^2=b^2k^2=b^2frac{Q'(t)^{2}}{Q(t)^{2}}. 130 end{equation} 131 于是我们就得到了微分方程 132 egin{equation} 133 label{eq:11.54} 134 (frac{P'(t)}{Q'(t)})^2+1=frac{b^2}{Q(t)^2}. 135 end{equation} 136 也就是 137 $$ 138 (frac{dP(t)}{dQ(t)})^2+1=frac{b^2}{Q(t)^2}. 139 $$ 140 也即 141 $$ 142 frac{dx}{dy}=-sqrt{(frac{b}{y})^2-1}. 143 $$ 144 令 $frac{b}{y}=cosh a$.其中 $ain mathbf{R}^{+}$,于是, 145 $$ 146 frac{dy}{da}=frac{-b anh a}{cosh a}. 147 $$ 148 且 149 $$ 150 frac{dx}{dy}=-sinh a. 151 $$ 152 因此, 153 $$ 154 frac{dx}{da}=b( anh a)^2=b-b anh'a. 155 $$ 156 因此, 157 $$ 158 x=ba-b anh a+C. 159 $$ 160 因此, 161 $$ 162 x=bcosh^{-1}frac{b}{y}-b anh(cosh^{-1}frac{b}{y})+C. 163 $$ 164 将初始条件 $x=0,y=b$ 代入,解得 $C=0$.于是 $B$ 的光滑轨迹为 165 $$ 166 x=bcosh^{-1}frac{b}{y}-b anh(cosh^{-1}frac{b}{y}). 167 $$ 168 通过这个方程,我们发现 $B$ 的运动轨迹和 $A$ 的运动无关!\ 169 170 当 $Q(t)=0$ 时,易得 $B$ 已经和 $A$ 同在 $x$ 轴上运动. 171 end{proof} 172 % ---------------------------------------------------------------------------------------- 173 % BIBLIOGRAPHY 174 % ---------------------------------------------------------------------------------------- 175 176 ibliographystyle{unsrt} 177 178 ibliography{sample} 179 180 % ---------------------------------------------------------------------------------------- 181 end{document}