高等数学—微分方程

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>&nbsp;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>

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>

形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="y'+p(x)y=0"><span></span><span></span></span>

形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="y'+p(x)y = q(x)"><span></span><span></span></span>

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>

令y'=p则y''=p' 带入原方程变成p'=f(x,p)一个一阶方程

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

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

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

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

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

<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>