From 33dcd5a9efc38fbeae6ebeaf7f39d13795865611 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Thu, 10 Jun 2021 23:14:20 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E5=BE=AE=E5=88=86=E6=96=B9?= =?UTF-8?q?=E7=A8=8B=E4=B8=8E=E7=9F=A9=E9=98=B5?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../differential-equation.pdf | Bin 95684 -> 100184 bytes .../differential-equation.tex | 39 +++++++++++++++++ linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 14355 -> 32966 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 41 ++++++++++++++++++ 4 files changed, 80 insertions(+) diff --git a/advanced-math/knowledge/9-differential-equation/differential-equation.pdf b/advanced-math/knowledge/9-differential-equation/differential-equation.pdf index 9840215170b548aa82d94d3efe727f367370ef9c..56071023e7cf2f83bbff3e3760c85bc647f6cf93 100644 GIT binary patch delta 74720 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a/advanced-math/knowledge/9-differential-equation/differential-equation.tex +++ b/advanced-math/knowledge/9-differential-equation/differential-equation.tex @@ -3,6 +3,7 @@ \usepackage{color} % 使用颜色 \definecolor{orange}{RGB}{255,127,0} +\definecolor{aqua}{RGB}{0,255,255} \usepackage{geometry} \setcounter{tocdepth}{4} \setcounter{secnumdepth}{4} @@ -222,4 +223,42 @@ $\dfrac{\textrm{d}p}{p}=\dfrac{2x}{1+x^2}\textrm{d}x$,$\ln p=\ln(1+x^2)+C'$, 所以综上$y=C_2e^{C_1y}$。 +\section{高阶线性微分方程} + +第一部分是一阶微分方程,分为可分离变量微分方程、齐次微分方程、一阶齐次线性微分方程、一阶非齐次线性微分方程。 + +第二部分是可降阶的高阶微分方程,分为三种。 + +第三部分就是本节的高阶线性微分方程,$y^{(n)}+a_1(x)y^{(n-1)}+\cdots+a_{n-1}(x)y'+a_n(x)y=0$就是$n$阶齐次线性微分方程,$y^{(n)}+a_1(x)y^{(n-1)}+\cdots+a_{n-1}(x)y'+a_n(x)y=f(x)$就是$n$阶非齐次线性微分方程 + +若$\varphi_1(x)$与$\varphi_2(x)$为两个函数,当$\varphi_1(x)$与$\varphi_2(x)$不成比例,则称$\varphi_1(x)$与$\varphi_2(x)$线性无关,否则$\varphi_1(x)$与$\varphi_2(x)$线性相关。 + +\textcolor{aqua}{\textbf{定理:}}若$\varphi_1(x)$与$\varphi_2(x)$为$y^{(n)}+a_1(x)y^{(n-1)}+\cdots+a_{n-1}(x)y'+a_n(x)y=0$的解,则$y=C_1\varphi_1(x)+C_2\varphi_2(x)$也为其解。 + +证明:因为$\varphi_1(x)$与$\varphi_2(x)$为解,所以代入方程: + +$\varphi_1''+a(x)\varphi_1'+b(x)\varphi_1=0$,$\varphi_2''+a(x)\varphi_2'+b(x)\varphi_2=0$ + +从而$(C_1\varphi_1+C_2\varphi_2)''+a(x)(C_1\varphi_1+C_2\varphi_2)'+b(x)(C_1\varphi_1+C_2\varphi_2)=C_1(\varphi_1''+a(x)\varphi_1'+b(x)\varphi_1)+C_2(\varphi_2''+a(x)\varphi_2'+b(x)\varphi_2)=0$。 + +所以得证。 + +\textcolor{aqua}{\textbf{定理:}}若$\varphi_1(x)$与$\varphi_2(x)$分别为$y^{(n)}+a_1(x)y^{(n-1)}+\cdots+a_{n-1}(x)y'+a_n(x)y=0$与$y^{(n)}+a_1(x)y^{(n-1)}+\cdots+a_{n-1}(x)y'+a_n(x)y=f(x)$的解,则$y=\varphi_1(x)+\varphi_2(x)$为$y^{(n)}+a_1(x)y^{(n-1)}+\cdots+a_{n-1}(x)y'+a_n(x)y=f(x)$的解。 + +证明:$\varphi_1''+a(x)\varphi_1'+b(x)\varphi_1=0$,$\varphi_2''+a(x)\varphi_2'+b(x)\varphi_2=f(x)$,代入$y=\varphi_1(x)+\varphi_2(x)$: + +$(\varphi_1+\varphi_2)''+a(x)(\varphi_1+\varphi_2)'+b(x)(\varphi_1+\varphi_2)=(\varphi_1''+a(x)\varphi_1'+b(x)\varphi_1)+(\varphi_2''+a(x)\varphi_2'+b(x)\varphi_2)=f(x)$。 + +所以得证。 + +\textcolor{aqua}{\textbf{定理:}}若$\varphi_1(x)$与$\varphi_2(x)$为$y^{(n)}+a_1(x)y^{(n-1)}+\cdots+a_{n-1}(x)y'+a_n(x)y=f(x)$的解,则$y=\varphi_1(x)-\varphi_2(x)$为$y^{(n)}+a_1(x)y^{(n-1)}+\cdots+a_{n-1}(x)y'+a_n(x)y=0$的解。 + +证明:$\varphi_1''+a(x)\varphi_1'+b(x)\varphi_1=f(x)$,$\varphi_2''+a(x)\varphi_2'+b(x)\varphi_2=f(x)$,代入$y=\varphi_1(x)-\varphi_2(x)$: + +$(\varphi_2-\varphi_1)''+a(x)(\varphi_2-\varphi_1)'+b(x)(\varphi_2-\varphi_1)$ + +$(\varphi_2''+a(x)\varphi_2'+b(x)\varphi_2)-(\varphi_1''+a(x)\varphi_1'+b(x)\varphi_1)$ + +$f(x)-f(x)=0$,所以得证。 + \end{document} diff --git a/linear-algebra/knowledge/2-matrix/matrix.pdf b/linear-algebra/knowledge/2-matrix/matrix.pdf index 0f0c2927e2302b4cff449dbb0786a28390effb52..76a04409b5bc56d51785b699983f975de6a70953 100644 GIT binary patch delta 28643 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n$个数$a_{ij}$(元素)排成的$m$行$n$列的数表。 + +元素是实数的矩阵称为实矩阵,元素是复数的矩阵是复矩阵。 + +行数列数都为$n$的就是$n$阶矩阵或方阵,记为$A_n$。 + +行矩阵或行向量:只有一行的矩阵$A=(a_1a_2\cdots a_n)$。 + +列矩阵或列向量:只有一列的矩阵$B= +\left(\begin{array}{c} + b_1 \\ + b_2 \\ + \cdots \\ + b_m +\end{array}\right)$。 + +同型矩阵:两个矩阵行数、列数相等。 + +相等矩阵:是同型矩阵,且对应元素相等的矩阵。记为$A=B$。 + +零矩阵:元素都是零的矩阵,记为$O$,但是不同型的零矩阵不相等。 + +\begin{multicols}{2} + + + 对角矩阵或对角阵:从左上角到右下角的直线(对角线)以外的元素都是0的矩阵,记为$\varLambda=\textrm{diag}(\lambda_1,\lambda_2,\cdots,\lambda_n)$。 + + $\varLambda=\left( + \begin{array}{cccc} + \lambda_1 & 0 & \cdots & 0 \\ + \end{array} + \right)$ + + 单位矩阵或单位阵:$\lambda_1=\lambda_2=\cdots=\lambda_n=1$的对角矩阵,记为$E$。 + + + +\end{multicols} + \end{document}