高等数学—微分方程
2021-05-20 18:00:57 83 举报AI智能生成
高等数学—微分方程公式全解
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大纲/内容
基本概念和一阶微分方程
微分方程的基本概念
微分方程：含有自变量、未知函数和未知函数的导数（或微分的方程）
微分方程的阶：微分方程中未知函数的导数的最高阶数为该微分方程的阶
微分方程的解、通解和特解：<br>带入微分方程能使方程成为恒等式的函数成为微分方程的解<br>通解：就是含有独立常数的个数与方程阶数相同的解（包含<span class="equation-text" data-index="0" data-equation="C_1" contenteditable="false"><span></span><span></span></span>,<span class="equation-text" contenteditable="false" data-index="1" data-equation="C_2,···C_n"><span></span><span></span></span>）<br>特解：不含有任意常数或任意常数确定的解（通解区别在于常数确定）
初始条件：要求自变量取某定值时，对应函数与各阶导数取指定的值，<br>这种条件称为初始条件，满足初始条件的解成为满足该初始条件的特解
线性方程：如果未知函数和他的各阶导数都是一次项，而且他们的系数只是自变量的函数或常数<br>则称这种微分方程为线性微分方程（每个y,<span class="equation-text" contenteditable="false" data-index="0" data-equation="y',y''"><span></span><span></span></span>···的次数为1 系数为常数或关于x的方程）
一阶方程及其解法
可分离变量
前提条件：形如<span class="equation-text" data-index="0" data-equation="y'=f(x)g(y),g(y)\neq0 " contenteditable="false"><span></span><span></span></span> 通解<span class="equation-text" contenteditable="false" data-index="1" data-equation="\int \frac{dy}{g(y)}=\int f(x) dx + c"><span></span><span></span></span>
齐次方程
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{dy}{dx} = f(\frac{y}{x})"><span></span><span></span></span>
<span style="font-size: inherit;">令<span class="equation-text" data-index="0" data-equation="\frac{y}{x}=u" contenteditable="false"><span></span><span></span></span></span><br>则<span class="equation-text" data-index="1" data-equation="\frac{dy}{dx} = u+\frac{du}{dx}=f(u)" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="2" data-equation="\int \frac{du}{f(u)-u}=\int \frac{dx}{x}+C = \ln |x| +C"><span></span><span></span></span><br>
一阶齐次线性方程
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="y'+p(x)y=0"><span></span><span></span></span>
通解公式：<span class="equation-text" contenteditable="false" data-index="0" data-equation="Ce^{-\int p(x)\mathrm{d}x}"><span></span><span></span></span>
一阶非齐次线性方程
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="y'+p(x)y = q(x)"><span></span><span></span></span>
通解公式：<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=e^{-\int p(x)\mathrm{d}x}(\int q(x)e^{\int p(x)\mathrm{d}x}\mathrm{d}x +C)"><span></span><span></span></span>
伯努利方程
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="y'+p(x)y=q(x)y^n"><span></span><span></span></span>
核心思想：两边同时除以<span class="equation-text" data-index="0" data-equation="y'" contenteditable="false"><span></span><span></span></span>再令<span class="equation-text" contenteditable="false" data-index="1" data-equation="z=y^{1-n}"><span></span><span></span></span>
通解公式：<span class="equation-text" contenteditable="false" data-index="0" data-equation="z=e^{-\int (1-n)p(x)\mathrm{d}x}(\int (1-n)q(x)e^{\int (1-n))p(x)\mathrm{d}x}\mathrm{d}x +C)(z=y^{1-n})"><span></span><span></span></span>
高阶微分方程
可降阶的二阶方程及其解法
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="y''=f(x,y')"><span></span><span></span></span>成为不含y的方程
令y'=p则y''=p' 带入原方程变成p'=f(x,p)一个一阶方程
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="y''=f(y,y')"><span></span><span></span></span>称为不含x的方程
令y'=p 则<span class="equation-text" data-index="0" data-equation="y''=p\frac{dp}{dy}" contenteditable="false"><span></span><span></span></span>带入原方程有<span class="equation-text" contenteditable="false" data-index="1" data-equation="p\frac{\mathrm{d}p}{\mathrm{d}y}=f(y,p)"><span></span><span></span></span>的一个一阶方程
高阶线性方程结构
二阶其次线性方程： <span class="equation-text" data-index="0" data-equation="y''+p(x)y'+q(x)y = 0" contenteditable="false"><span></span><span></span></span><br>二阶非齐次线性方程：<span class="equation-text" contenteditable="false" data-index="1" data-equation="y''+p(x)y'+q(x)y = f(x)"><span></span><span></span></span>
若<span class="equation-text" data-index="0" data-equation="y_1(x),y_2(x)" contenteditable="false"><span></span><span></span></span>为二阶齐次方程的两个特解，<br>则他们的线性组合仍为同方程的解，<br>特别的，当<span class="equation-text" data-index="1" data-equation="y_1(x)\neq \lambda y_2(x)(\lambda为常数)" contenteditable="false"><span></span><span></span></span>，也即<span class="equation-text" data-index="2" data-equation="y_1(x)与y_2(x)" contenteditable="false"><span></span><span></span></span>的线性无关时，<br>则方程通解为<span class="equation-text" contenteditable="false" data-index="3" data-equation="y=C_1 y_1(x) + y_2(x)"><span></span><span></span></span>
若<span class="equation-text" data-index="0" data-equation="y_1(x),y_2(x)" contenteditable="false"><span></span><span></span></span>为二阶非齐次线性方程的两个特解，<br>则<span class="equation-text" contenteditable="false" data-index="1" data-equation="y_1(x)-y_2(x)"><span></span><span></span></span>为对应的二阶齐次方程的一个特解
若<span class="equation-text" data-index="0" data-equation="\overline{y}(x)" contenteditable="false"><span></span><span></span></span>为二阶非齐次方程的一个特解，而<span class="equation-text" data-index="1" data-equation="y(x)" contenteditable="false"><span></span><span></span></span>为对应的二阶齐次方程的任意特解，<br>则<span class="equation-text" contenteditable="false" data-index="2" data-equation="\overline y(x)+y(x)"><span></span><span></span></span>为此二阶非齐次方程的一个特解
若<span class="equation-text" data-index="0" data-equation="\overline{y}(x)" contenteditable="false"><span></span><span></span></span>为二阶非齐次方程的一个特解，而<span class="equation-text" data-index="1" data-equation="C_1 y_1(x)+C_2 y_2(x)" contenteditable="false"><span></span><span></span></span>为对应的二阶齐次方程的任意特解，<br>则<span class="equation-text" contenteditable="false" data-index="2" data-equation="\overline{y}(x)+C_1 y_1(x)+C_2 y_2(x)"><span></span><span></span></span>为此二阶非齐次方程的一个特解
<b>解的叠加原理</b>：<br>设<span class="equation-text" data-index="0" data-equation="y_1(x)" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" data-index="1" data-equation="y_2(x)" contenteditable="false"><span></span><span></span></span>分别时<span class="equation-text" data-index="2" data-equation="y''+p(x)y'+q(x)y=f_1(x)" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" data-index="3" data-equation="y''+p(x)y'+q(x)y=f_2(x)" contenteditable="false"><span></span><span></span></span>的特解<br>则<span class="equation-text" data-index="4" data-equation="y_1(x)+y_2(x)" contenteditable="false"><span></span><span></span></span>是<span class="equation-text" contenteditable="false" data-index="5" data-equation="y''+p(x)y'+q(x)y=f_1(x)+f_2(x)"><span></span><span></span></span>的特解
二阶、高阶常系数齐次线性方程及其解法
二阶常系数齐次线性方程<br><span class="equation-text" contenteditable="false" data-index="0" data-equation="y''+py'+qy=0"><span></span><span></span></span>(p.q为常数)
特征方程：<span class="equation-text" data-index="0" data-equation="\lambda^2 + p\lambda +q = 0" contenteditable="false"><span></span><span></span></span><br>特征方程跟的三种不同情形（<span class="equation-text" contenteditable="false" data-index="1" data-equation="\delta = p^2-4q"><span></span><span></span></span>）对应方程通解的三种形式
<span class="equation-text" data-index="0" data-equation="\delta >0" contenteditable="false"><span></span><span></span></span>，特征方程有两个不同的实根<span class="equation-text" contenteditable="false" data-index="1" data-equation="\lambda_1,\lambda_2"><span></span><span></span></span><br>方程通解为：<span class="equation-text" data-index="2" data-equation="y(x)=C_1 e^{\lambda_1x}+C_2 e^{\lambda_2x}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\delta =0" contenteditable="false"><span></span><span></span></span>，特征方程有两重根<span class="equation-text" contenteditable="false" data-index="1" data-equation="\lambda"><span></span><span></span></span><br>方程通解为：<span class="equation-text" data-index="2" data-equation="y(x)=（C_1+C_2 x) e^{\lambda x}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\delta <0" contenteditable="false"><span></span><span></span></span>，特征方程有共轭复根<span class="equation-text" data-index="1" data-equation="\lambda_1=\alpha+\beta i,\lambda_2=\alpha-\beta i" contenteditable="false"><span></span><span></span></span><br>方程通解为：<span class="equation-text" data-index="2" data-equation="y(x)=e^{\alpha x}(C_1\cos{\beta x}+C_2\sin{\beta x})" contenteditable="false"><span></span><span></span></span><br>(证明过程：欧拉公式：<span class="equation-text" contenteditable="false" data-index="3" data-equation="e^{\alpha \pm \beta i} = e^{\alpha }(\cos{\beta}\pm i\sin{\beta})"><span></span><span></span></span>)
二阶，高阶常系数非齐次线性方程及其解法
二阶常系数齐次线性方程<br><span class="equation-text" contenteditable="false" data-index="0" data-equation="y''+py'+qy=f(x)"><span></span><span></span></span>(p.q为常数)
先求齐次方程通解
特解形式：（不包含三角函数）<br><span class="equation-text" data-index="0" data-equation="f(x) = p_n(x)e^{\alpha x}" contenteditable="false"><span></span><span></span></span>，其中<span class="equation-text" contenteditable="false" data-index="1" data-equation="p_n(x)"><span></span><span></span></span>为n次多项式，<span class="equation-text" data-index="2" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>为常数
若<span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>不是特征根，则令<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overline{y(x)} = Q_n(x)e^{\alpha x}"><span></span><span></span></span>
若<span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>是特征单根，则令<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overline{y(x)} = xQ_n(x)e^{\alpha x}"><span></span><span></span></span>
若<span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>是二重根，则令<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overline{y(x)} = x^2Q_n(x)e^{\alpha x}"><span></span><span></span></span>
特解形式：（包含三角函数）<br><span class="equation-text" data-index="0" data-equation="f(x) = e^{\alpha x}(P_l\cos{\beta x} + P_m\sin{\beta x})" contenteditable="false"><span></span><span></span></span><br>其中<span class="equation-text" contenteditable="false" data-index="1" data-equation="P_l(x)"><span></span><span></span></span>为<span class="equation-text" data-index="2" data-equation="l" contenteditable="false"><span></span><span></span></span>次多项式
若<span class="equation-text" data-index="0" data-equation="\alpha \pm i\beta" contenteditable="false"><span></span><span></span></span> 不是特征根<br>则令<span class="equation-text" data-index="1" data-equation="\overline{y}=e^{\alpha x}({R_n(x)\cos{\beta x}+T_n(x)\sin{\beta x}})" contenteditable="false"><span></span><span></span></span><br>其中<span class="equation-text" contenteditable="false" data-index="2" data-equation="n=max(l,m),R_n(x),T_n(x)"><span></span><span></span></span>为两个n次多项式
若<span class="equation-text" data-index="0" data-equation="\alpha \pm i\beta" contenteditable="false"><span></span><span></span></span> 是特征根<br>则令<span class="equation-text" data-index="1" data-equation="\overline{y}=xe^{\alpha x}({R_n(x)\cos{\beta x}+T_n(x)\sin{\beta x}})" contenteditable="false"><span></span><span></span></span><br>其中<span class="equation-text" contenteditable="false" data-index="2" data-equation="n=max(l,m),R_n(x),T_n(x)"><span></span><span></span></span>为两个n次多项式
欧拉方程及其解法
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^2 y''+p_1xy'+p_2y=q(x)"><span></span><span></span></span>为二阶欧拉方程
<b>令x=e^t<br><span class="equation-text" data-index="0" data-equation="y''=\frac {\mathrm{d} ^ 2y}{\mathrm{d}x^2}=-\frac{1}{x^2}\frac{\mathrm{d}y}{\mathrm{d}t}+\frac{1}{x^2}\frac{mathrm{d}^2y}{\mathrm{d}t^2}" contenteditable="false"><span></span><span></span></span><br></b>带入方程，变为t是自变量，y是未知函数的微分方程即<br><span class="equation-text" contenteditable="false" data-index="1" data-equation="y''(t)+(p_1-1)y'(t)+p_2y(e^t)=q(e^t)"><span></span><span></span></span><br>
线性相关：两个函数<span class="equation-text" data-index="0" data-equation="y_1(x)与y_2(x)" contenteditable="false"><span></span><span></span></span><br>若<span class="equation-text" data-index="1" data-equation="\frac{y_1(x)}{y_2(x)}=C (C为不关于x的函数)" contenteditable="false"><span></span><span></span></span>则<span class="equation-text" contenteditable="false" data-index="2" data-equation="y_1(x),y_2(x)"><span></span><span></span></span>称为线性相关<br>反之为线性无关
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