symmetry methods for differential equations,exercise 1.4
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do not edit title 50 % and author name here, see the TITLE block 51 % below 52 \renewcommand\refname{参考文献} 53 \newcommand{\D}{\displaystyle}\newcommand{\ri}{\Rightarrow} 54 \newcommand{\ds}{\displaystyle} \renewcommand{\ni}{\noindent} 55 \newcommand{\pa}{\partial} \newcommand{\Om}{\Omega} 56 \newcommand{\om}{\omega} \newcommand{\sik}{\sum_{i=1}^k} 57 \newcommand{\vov}{\Vert\omega\Vert} \newcommand{\Umy}{U_{\mu_i,y^i}} 58 \newcommand{\lamns}{\lambda_n^{^{\scriptstyle\sigma}}} 59 \newcommand{\chiomn}{\chi_{_{\Omega_n}}} 60 \newcommand{\ullim}{\underline{\lim}} \newcommand{\bsy}{\boldsymbol} 61 \newcommand{\mvb}{\mathversion{bold}} \newcommand{\la}{\lambda} 62 \newcommand{\La}{\Lambda} \newcommand{\va}{\varepsilon} 63 \newcommand{\be}{\beta} \newcommand{\al}{\alpha} 64 \newcommand{\dis}{\displaystyle} \newcommand{\R}{{\mathbb R}} 65 \newcommand{\N}{{\mathbb N}} \newcommand{\cF}{{\mathcal F}} 66 \newcommand{\gB}{{\mathfrak B}} \newcommand{\eps}{\epsilon} 67 \begin{flushright} % Right align 68 {\LARGE\@title} % Increase the font size of the title 69 70 \vspace{50pt} % Some vertical space between the title and author name 71 72 {\large\@author} % Author name 73 \\\@date % Date 74 75 \vspace{40pt} % Some vertical space between the author block and abstract 76 \end{flushright} 77 } 78 79 % ---------------------------------------------------------------------------------------- 80 % TITLE 81 % ---------------------------------------------------------------------------------------- 82 \begin{document} 83 \begin{CJK}{UTF8}{gkai} 84 \title{\textbf{Symmetry Methods for Differential Equations:\\Exercise 1.4}} 85 % \setlength\epigraphwidth{0.7\linewidth} 86 \author{\small{叶卢庆}\\{\small{杭州师范大学理学院,学号:1002011005}}\\{\small{Email:h5411167@gmail.com}}} % Institution 87 \renewcommand{\today}{\number\year. \number\month. \number\day} 88 \date{\today} % Date 89 90 % ---------------------------------------------------------------------------------------- 91 92 93 \maketitle % Print the title section 94 95 % ---------------------------------------------------------------------------------------- 96 % ABSTRACT AND KEYWORDS 97 % ---------------------------------------------------------------------------------------- 98 99 % \renewcommand{\abstractname}{摘要} % Uncomment to change the name of the abstract to something else 100 101 % \begin{abstract} 102 103 % \end{abstract} 104 105 % \hspace*{3,6mm}\textit{关键词:} % Keywords 106 107 % \vspace{30pt} % Some vertical space between the abstract and first section 108 109 % ---------------------------------------------------------------------------------------- 110 % ESSAY BODY 111 % ---------------------------------------------------------------------------------------- 112 \begin{exercise}[1.4] 113 Determine the value of $\alpha$ for which 114 $$ 115 (x',y')=(x+2\va,ye^{\alpha\va}) 116 $$ 117 is a symmetry of 118 $$ 119 \frac{dy}{dx}=y^2e^{-x}+y+e^x 120 $$ 121 for all $\va\in\mathbf{R}$. 122 \end{exercise} 123 \begin{proof} 124 The symmetry condition for the differential equation is 125 $$ 126 \frac{\frac{\pa g}{\pa x}+\frac{\pa g}{\pa y}w(x,y)}{\frac{\pa f}{\pa 127 x}+\frac{\pa f}{\pa y}w(x,y)}=w(f(x,y),g(x,y)). 128 $$ 129 Where 130 $w(x,y)=y^2e^{-x}+y+e^x$,$f(x,y)=x+2\va,g(x,y)=ye^{\alpha\va}$.So the 131 symmetry condition can be written as 132 $$ 133 y^2e^{-x+\alpha\va}+e^{x+\alpha\va}=y^2e^{2\alpha\va-x-2\va}+e^{x+2\va}. 134 $$ 135 So $\alpha=2$. 136 \end{proof} 137 % ---------------------------------------------------------------------------------------- 138 % BIBLIOGRAPHY 139 % ---------------------------------------------------------------------------------------- 140 141 \bibliographystyle{unsrt} 142 143 \bibliography{sample} 144 145 % ---------------------------------------------------------------------------------------- 146 \end{CJK} 147 \end{document}