%我的 tex 模版 documentclass[UTF8,a1paper,landscape]{ctexart}%UTF8 中文支持,a1paper 纸张大小,landscape 横向版面,ctexart 中文文章 usepackage{tikz}%图包 usetikzlibrary{trees}%树包 usepackage{amsmath} usepackage{geometry}%页边距设置 geometry{top=5cm,bottom=5cm,left=5cm,right=5cm} usepackage{fancyhdr}%页头页尾页码设置 pagestyle{fancy} egin{document} itle{常微分方程图解} author{dengchaohai} maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ewpage%另起一页 part{一阶常微分方程} section*{一阶常微分方程逻辑关系图解} egin{center}%居中 egin{tikzpicture} [ grow=right, r/.style={rectangle,draw,fill=red!20,align=center,rounded corners=.8ex}, g/.style={rectangle,draw,fill=green!20,align=center,rounded corners=.8ex}, b/.style={rectangle,draw,fill=blue!20,align=center,rounded corners=.8ex}, grow via three points={one child at (4,-4) and two children at (4,-4) and (4,-8)}, edge from parent path={( ikzparentnode.south)|-( ikzchildnode.west)}, ]%属性定义 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ode(-1)at(0,12)[r]{解的存在唯一性\$/x-x_0/leq h$} child{node(g)[g]{解的延拓}} child{node(h)[g]{解对初值的连续可微性}} child[missing]{} child[missing]{} child[missing]{} child[missing]{}; ode(-2)at(10,8)[b]{解的存在空间（局部）\$|x-x_0|leq h$}; ode(-3)at(15,8)[b]{通解\$y=y(c,x)$}; ode(-4)at(20,8)[b]{定解\$y=y(x_0,y_0,x)$}; ode(-5)at(25,8)[b]{解的存在空间（饱和）\$(c,d)$}; ode(-6)at(15,0)[b]{包络}; ode(-7)at(20,4)[b]{解对初值的连续可微性\$y=y(x_0,y_0,x,lambda)$}; ode(-8)at(40,0)[b]{奇解} child{node[b]{$c-$判别曲线\$Phi(c,x,y)=0,Phi'_c=0$}} child{node[b]{$p-$判别曲线\$F(x,y,p)=0,F'_p=0$}}; draw[->](-2)--(-3); draw[->](-3)--(-6); draw[->](-4)--(-7); draw[->](-3)to node[above]{初值$(x_0,y_0)$}(-4); draw[->](-4)--(-5); draw[->](-2)--(10,9)to node[above]{延拓}(25,9)--(-5); draw[->](-6)--(-8); ode(0)at(0,0)[r]{一阶常微分方程\$F(x,y,y')=0$} child{node[g]{显式\$y'=f(x,y)$} child{node(a)[b]{3分式微分方程\$frac{dy}{dx}=frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}$}} child{node(b)[b]{7伯努利微分方程\$frac{dy}{dx}=P(x)y+Q(x)y^n$}}} child[missing]{} child[missing]{} child{node[g]{隐式\$F(x,y,y')=0$} child{node(c)[b]{8显解$x$\$x=f(y,y')$}} child{node(d)[b]{9显解$y$\$y=f(x,y')$}} child{node(e)[b]{10不含$x$\$F(y,y')=0$}} child{node(f)[b]{11不含$y$\$F(x,y')=0$}}}; ode(1)at(15,-8)[b]{2齐次微分方程\$frac{dy}{dx}=f(frac{y}{x})$}; ode(2)at(20,-8)[b]{1变量分离方程\$frac{dy}{dx}=psi(x)varphi(y)$}; ode(3)at(25,-8)[b]{4恰当微分方程\$M(x,y)dx+N(x,y)dy=0$}; ode(4)at(15,-12)[b]{6非齐次线性微分方程\$frac{dy}{dx}=P(x)y+Q(x)$}; ode(5)at(20,-12)[b]{5齐次线性微分方程\$frac{dy}{dx}=P(x)y$}; draw[->](a)--(1); draw[->](1)--(2); draw[->](2)--(3); draw[->](b)--(4); draw[->](4)--(5); draw[->](5)--(2); draw(-1)--(0); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end{tikzpicture} end{center} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ewpage section*{一阶常微分方程图解对应解法} egin{itemize} item 1. [ egin{split} given:quadfrac{dy}{dx}=psi(x)varphi(y).\ ifquadvarphi(y)=0\ &Rightarrow varphi(y)=0\ &Rightarrow y=y_0\ ifquadvarphi(y) e0\ &Rightarrow frac{1}{varphi(y)}dy=psi(x)dx\ &Rightarrow intfrac{1}{varphi(y)}dy=intpsi(x)dx+c\ &Rightarrow Phi(c,x,y)=0\ &Rightarrow y=y(c,x)\ end{split} ] item 2. [ egin{split} given:quadfrac{dy}{dx}=f(frac{y}{x}).\ letquad u=frac{y}{x} &Rightarrow y=ux\ &Rightarrow frac{dy}{dx}=frac{du}{dx}x+u\ &Rightarrow frac{du}{dx}x+u=f(u)\ &Rightarrow frac{du}{dx}=frac{f(u)-u}{x}\ &Rightarrow frac{du}{dx}=psi(u)varphi(x)\ end{split} ] item 3. [ egin{split} given:quadfrac{dy}{dx}=frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}.\ ifquad c_1=c_2=0\ &Rightarrow frac{dy}{dx}=frac{a_1+b_1frac{y}{x}}{a_2+b_2frac{y}{x}}\ &Rightarrow frac{dy}{dx}=f(frac{y}{x})\ letquad u=frac{y}{x}\ &Rightarrow frac{dy}{dx}=psi(u)\ ifquad frac{a_1}{a_2}=frac{b_1}{b_2}=frac{c_1}{c_2}=k\ &Rightarrow frac{dy}{dx}=k\ ifquad frac{a_1}{a_2}=frac{b_1}{b_2}=k efrac{c_1}{c_2}\ letquad u=a_1x+b_1y\ &Rightarrow frac{du}{dx}=a_1+b_1frac{dy}{dx}=a_1+b_1frac{ku+c_1}{u+c_2}\ &Rightarrow frac{du}{dx}=psi(u)\ ifquad frac{a_1}{a_2} efrac{b_1}{b_2}\ &Rightarrow a_1x+b_1y+c_1=0,a_2x+b_2y+c_2=0\ &Rightarrow x=x_0,y=y_0\ letquad x=X+x_0,y=Y+y_0\ &Rightarrow frac{dy}{dx}=frac{dY}{dX}=frac{a_1X+b_1Y}{a_2X+b_2X}=frac{a_1+b_1frac{Y}{X}}{a_2+b_2frac{Y}{X}}\ letquad u=frac{Y}{X}\ &Rightarrow frac{du}{dX}=psi(u)varphi(X)\ end{split} ] item 4. [ egin{split} given:quad M(x,y)dx+N(x,y)dy=0.\ ifquad frac{frac{partial M}{partial y}-frac{partial N}{partial x}}{-M}=varphi(y)\ &Rightarrow mu=e^{intvarphi(y)dy}\ ifquad frac{frac{partial M}{partial y}-frac{partial N}{partial x}}{N}=psi(x)\ &Rightarrow mu=e^{intvarphi(x)dx}\ letquad mu Mdx+mu Ndy=0\ &Rightarrow u=intmu Mdx+varphi(y)\ &Rightarrow frac{du}{dy}=mu N\ &Rightarrow varphi(y)\ &Rightarrow u=Phi(x,y)\ &Rightarrow Phi(x,y)=c\ end{split} ] item 5. [ egin{split} given:quadfrac{dy}{dx}=P(x)y.\ &Rightarrow y=ce^{int P(x)dx}\ end{split} ] item 6. [ egin{split} given:quadfrac{dy}{dx}=P(x)y+Q(x).\ &Rightarrow y=e^{int P(x)dx}(int Q(x)e^{-int P(x)dx}dx+c)\ end{split} ] item 7. [ egin{split} given:quadfrac{dy}{dx}=P(x)y+Q(x)y^n.\ y^{-n}\ &Rightarrow y^{-n}frac{dy}{dx}=y^{-n}P(x)y+y^{-n}Q(x)y^n\ &Rightarrow y^{-n}frac{dy}{dx}=P(x)y^{1-n}+Q(x)\ letquad z=y^{1-n}\ &Rightarrow frac{dz}{dx}=(1-n)y^{-n}frac{dy}{dx}\ &Rightarrow frac{dz}{dx}=(1-n)P(x)z+(1-n)Q(x)\ &Rightarrow frac{dz}{dx}=psi (x)z+varphi(x)\ end{split} ] item 8. item 9. [ egin{split} given:quad y=f(x,y').\ letquad p=y'\ &Rightarrow y=f(x,p)\ &Rightarrow frac{dy}{dx}=frac{partial f}{partial x}+frac{partial f}{partial p}frac{dp}{dx}\ &Rightarrow p=varphi(c,x)\ &Rightarrow x=psi(c,p)\ &Rightarrow y=f(psi(c,p),p)\ end{split} ] item 10. item 11. [ egin{split} given:quad F(x,y')=0.\ letquad p=y'=p(t,x)\ &Rightarrow x=psi(t)\ &Rightarrow p=varphi(t)\ &Rightarrow dy=pdx=varphi(t)psi'(t)dt\ &Rightarrow y=intvarphi(t)psi'(t)dt+c\ end{split} ] end{itemize} ewpage part{高阶常微分方程} section*{高阶常微分方程逻辑关系图解} egin{center}%居中 egin{tikzpicture} [ grow=right, r/.style={rectangle,draw,fill=red!20,align=center,rounded corners=.8ex}, g/.style={rectangle,draw,fill=green!20,align=center,rounded corners=.8ex}, b/.style={rectangle,draw,fill=blue!20,align=center,rounded corners=.8ex}, grow via three points={one child at (4,-2) and two children at (4,-2) and (4,-4)}, edge from parent path={( ikzparentnode.south)|-( ikzchildnode.west)}, ]%属性定义 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ode(0)at(0,0)[r]{高阶常微分方程} child[missing]{} child[missing]{} child{node(a)[g]{齐次微分方程} child{node[b]{常系数线性齐次微分方程} child{node[b]{特征根法求通解} child{node[b]{单根} child{node[b]{实单根\$e^{lambda_1t}$}} child{node[b]{$e^{lambda t}=e^{alpha t}(cos(eta t)+sin(eta t))$\复单根\(复$=2$,单$=1Rightarrow 2 imes1$的矩阵)\$e^{lambda_1t}$\$e^{overline{lambda_1}t}$}}} child[missing]{} child[missing]{} child{node[b]{重根} child{node[b]{实$k$重根\$underbrace{e^{lambda_1t},te^{lambda_1t},dots,t^{k-1}e^{lambda_1t}}_k$}} child[missing]{} child{node[b]{$e^{lambda t}=e^{alpha t}(cos(eta t)+sin(eta t))$\复$k$重根\(复$=2$,单$=kRightarrow 2 imes k$的矩阵)\$underbrace{e^{lambda_1t},te^{lambda_1t},dots,t^{k-1}e^{lambda_1t}}_k$\$underbrace{e^{overline{lambda_1}t},te^{overline{lambda_1}t},dots,t^{k-1}e^{overline{lambda_1}t}}_k$}}}}}} child[missing]{} child[missing]{} child[missing]{} child[missing]{} child[missing]{} child[missing]{} child[missing]{} child[missing]{} child[missing]{} child[missing]{} child{node(b)[g]{非齐次微分方程} child{node[b]{常系数非线性齐次微分方程} child{node[b]{待定系数法求特解} child{node[b]{$f(t)=(b_0t^m+b_1t^{m-1}+dots+b_m)e^{lambda t}$} child{node[b]{$k$是特征值$lambda$的重数,若$lambda$不是特征值,那么$k=0$\~{x}$=t^k((b_0t^m+b_1t^{m-1}+dots+b_m)e^{lambda t})$}}} child[missing]{} child[missing]{} child{node[b]{$f(t)=((P(t)cos(eta t)+Q(t)sin(eta t))e^{alpha t}$} child{node[b]{$e^{lambda t}=e^{alpha t}(cos(eta t)+sin(eta t))$\$k$是特征值$lambda$的重数,若$lambda$不是特征值,那么$k=0$\~{x}$=t^k(((P(t)cos(eta t)+Q(t)sin(eta t))e^{alpha t})$}}}}}}; ode(1)at(4,0)[g]{降阶法\$x=x(t)$} child{node[b]{不显含$x$,降$k$阶\$F(t,x^{(k)},x^{(k+1)},dots,x^{(n)})=0$}} child{node[b]{不显含$t$,降$1$阶\$F(x,x',x'',dots,x^{(n)})=0$}}; draw[->](0)--(1); draw[->,green](a)to node[right]{常数变易法$cRightarrow c(x)$}(b); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end{tikzpicture} end{center} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ewpage section*{高阶常微分方程图解对应解法} egin{itemize} item 1.降$1$阶 [ egin{split} givenquad F(x,x',x'',dots,x^{(n)})=0.\ letquad y=x'\ &Rightarrow x''=frac{dy}{dt}=frac{dy}{dx}frac{dx}{dt}=y'frac{dy}{dx}\ &Rightarrow F(x,y,y'frac{dy}{dx},dots,cdotsfrac{dy}{dx})=0\ &Rightarrow y=y(t,c_1,c_2,dots,c_{n-1})\ &Rightarrow int ydx=int x'dx\ &Rightarrow x\ end{split} ] item 2.降$k$阶 [ egin{split} givenquad F(t,x^{(k)},x^{(k+1)},dots,x^{(n)})=0.\ letquad y=x^{(k)}\ &Rightarrow F(t,y,y',dots,y^{(n-k)})=0\ &Rightarrow y=y(t,c_1,c_2,dots,c_{n-k})\ &Rightarrow int ydx=int x^{(k)}dx\ &Rightarrow x=underbrace{idotsint}_k x^{(k)}dx\ end{split} ] end{itemize} ewpage part{常微分方程组} section*{常微分方程组逻辑关系图解} egin{center}%居中 egin{tikzpicture} [ grow=right, r/.style={rectangle,draw,fill=red!20,align=center,rounded corners=.8ex}, g/.style={rectangle,draw,fill=green!20,align=center,rounded corners=.8ex}, b/.style={rectangle,draw,fill=blue!20,align=center,rounded corners=.8ex}, grow via three points={one child at (4,-2) and two children at (4,-2) and (4,-4)}, edge from parent path={( ikzparentnode.south)|-( ikzchildnode.west)}, ]%属性定义 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 123 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end{tikzpicture} end{center} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end{document}