% -*- fill-column: 120 -*- % % Cheatsheet for Differential Equations % \documentclass[10pt]{article} \usepackage{fullpage} \usepackage{times} \usepackage{amsmath} \setlength\topmargin{-0.3in} %% \setlength\headheight{0in} %% \setlength\headsep{0in} \setlength\textheight{10.5in} \setlength\textwidth{6.5in} \setlength\hoffset{-0.2in} \setlength{\parindent}{0in} \pagestyle{empty} \thispagestyle{empty} \pagestyle{empty} \newcommand{\interspace}{\vspace{5mm}} \newcommand{\interboxsep}{\vspace{2mm}} \newcommand{\soln}{\textit{Sol}} \newcommand{\twocolwidth}{0.96\textwidth} \newcommand{\tightitems}{%% \setlength{\topsep}{0pt}%% \setlength{\itemsep}{0pt}%% \setlength{\parsep}{0pt}%% \setlength{\parskip}{0pt}%% } \newcommand{\casehead}[1]{\textit{\underline{#1}}} \newcommand{\TODO}{\emph{FIXME TODO}} \title{Differential Equations Cheatsheet} \author{} \date{} \begin{document} \thispagestyle{empty} \begin{center} {\LARGE Differential Equations Cheatsheet} \end{center} %------------------------------------------------------------------------------------------------------------- \section*{Jargon} {\small \textit{General Solution}: a family of functions, has parameters. \\ \textit{Particular Solution}: has no arbitrary parameters. \\ \textit{Singular Solution}: cannot be obtained from the general solution. } %------------------------------------------------------------------------------------------------------------- \section*{Linear Equations} \begin{center} $y^{(n)}(x) + a_{n-1}(x)y^{(n-1)}(x) + \dots + a_{1}(x)y'(x) + a_{0}(x)y(x) = f(x)$ \end{center} \subsection*{1st-order} \begin{center} {$F(y', y, x) = 0$ \qquad $y' + a(x)y = f(x)$ \qquad I.F. = $e^{\int a(x) dx}$ \qquad \soln: $y = C e^{-\int a(x) dx}$} \end{center} \interspace \begin{tabular}{cc} \begin{minipage}[t]{0.5\textwidth} % Left column \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Variable Separable} \begin{displaymath} \dfrac{dy}{dx} = f(x,y) \qquad A(x)dx + B(y)dy = 0 \end{displaymath} Test: \begin{displaymath} f(x,y)f_{xy}(x,y) = f_{x}(x,y)f_{y}(x,y) \end{displaymath} \soln: Separate and integrate on both sides. \end{minipage}} \interboxsep \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Exact} \begin{displaymath} M(x,y)dx + N(x,y)dy = 0 = dg(x,y) \end{displaymath} \begin{displaymath} \textrm{Iff}\qquad \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \end{displaymath} \soln: Find $g(x,y)$ by integrating and comparing: \begin{displaymath} \int M dx \qquad\textrm{and}\qquad \int N dy \end{displaymath} \end{minipage}} \interboxsep \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Reduction to Exact via Integrating Factor} \begin{displaymath} I(x,y)[M(x,y)dx + N(x,y)dy] = 0 \end{displaymath} \casehead{Case I} \vspace{1mm} \\ If \; $\dfrac{M_y - N_x}{M} \equiv h(y)$ \quad then \quad $I(x,y) = e^{-\int h(y)dx}$ \vspace{3mm} \casehead{Case II} \vspace{1mm} \\ If \; $\dfrac{N_x - M_y}{N} \equiv g(x)$ \quad then \quad $I(x,y) = e^{-\int g(x)dx}$ \vspace{3mm} \casehead{Case III} \vspace{1mm} \\ If $M = y f(xy)$ and $N = x g(xy)$ then $I(x,y) = \frac{1}{xM - yN}$ \end{minipage}} \end{minipage} \begin{minipage}[t]{0.5\textwidth} % Right column \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Homogeneous of degree 0} \begin{displaymath} f(tx,ty) = t^0f(x,y) = f(x,y) \end{displaymath} \soln: Reduce to var.sep. using: \begin{displaymath} y = xv \qquad \dfrac{dy}{dx} = v + x\frac{dv}{dx} \end{displaymath} \end{minipage}} \interboxsep \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Bernoulli} \begin{displaymath} y' + p(x)y = q(x)y^n \end{displaymath} \soln: Change var $z = \dfrac{1}{y^{n-1}}$ and divide by $\dfrac{1}{y^{n}}$. \end{minipage}} \interboxsep \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Reduction by Translation} \begin{displaymath} y' = \dfrac{Ax + By + C}{Dx + Ey + F} \end{displaymath} \casehead{Case I: Lines intersect} \vspace{1mm} \\ \soln: Put $x = X + h$ and $y = Y + k$, \\ find $h$ and $k$, solve var.sep. and translate back. \vspace{2mm} \casehead{Case II: Parallel Lines} ($A = B, D = E$) \vspace{1mm} \\ \soln: Put $u = Ax + By$,\; $y' = \dfrac{u' - A}{B}$ and solve. \end{minipage}} \end{minipage} \end{tabular} \vspace{1em} \subsection*{Principle of Superposition} If \; \parbox{2.3in}{ $y'' + ay' + by = f_1(x)$ \; has solution $y_1(x)$ \\ $y'' + ay' + by = f_2(x)$ \; has solution $y_2(x)$} \; then \; \parbox{2.2in}{ $y'' + ay' + by = f(x) = f_1(x) + f_2(x)$ \\ has solution: $ y(x) = y_1(x) + y_2(x)$} \clearpage %--------------------------------------------------------------------------------------------------- \subsection*{2nd-order Homogeneous} \begin{center} {$F(y'', y', y, x) = 0$ \qquad $y'' + a(x)y' + b(x)y = 0$ \qquad \soln: $y_h = c_1y_1(x) + c_2y_2(x)$} \end{center} \interspace \begin{tabular}{cc} \begin{minipage}{0.5\textwidth} % Left column \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Reduction of Order - Method} If we already know $y_1$, put $y_2 = vy_1$, \\ expand in terms of $v'', v', v$, and put $z = v'$ \\ and solve the reduced equation. \end{minipage}} \interboxsep \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Wronskian (Linear Independence)} $y_1(x)$ and $y_2(x)$ are linearly independent iff \begin{displaymath} W(y_1, y_2)(x) = \left| \begin{array}{cccc} y_1 & y_2 \\ y'_1 & y'_2 \end{array} \right| \neq 0 \end{displaymath} \end{minipage}} \end{minipage} \framebox{\begin{minipage}{0.5\textwidth} % Right column \subsubsection*{Constant Coefficients} \begin{displaymath} \textrm{A.E.} \qquad \lambda^2 + a\lambda + b = 0 \end{displaymath} \casehead{A. Real roots} \vspace{1mm} \\ \soln: $y(x) = C_1e^{\lambda_1x} + C_2e^{\lambda_2x}$ \vspace{2mm} \casehead{B. Single root} \vspace{1mm} \\ \soln: $y(x) = C_1e^{\lambda x} + C_2xe^{\lambda x}$ \vspace{2mm} \casehead{C. Complex roots} \vspace{1mm} \\ \soln: $y(x) = e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$ \\ with $\alpha = -\frac{a}{2}$ and $\beta = \frac{\sqrt{4b - a^2}}{2}$ \end{minipage}} \end{tabular} \interboxsep \framebox{ \begin{tabular}{cc} \begin{minipage}{0.5\textwidth} % Left column \subsubsection*{Euler-Cauchy Equation} \begin{displaymath} x^2y'' + axy' + by = 0\quad \textrm{where}\; x \neq 0 \end{displaymath} \begin{displaymath} A.E.: \quad \lambda(\lambda-1) + a\lambda + b = 0 \end{displaymath} \soln: $y(x)$ of the form $x^\lambda$ \textit{Reduction to Constant Coefficients:} Use $x = e^t, t = \ln{x}$, \\ and rewrite in terms of $t$ using the chain rule. \end{minipage} \begin{minipage}{0.5\textwidth} % Rightcolumn \vspace{2mm} \casehead{A. Real roots} \vspace{1mm} \\ \soln: $y(x) = C_1x^{\lambda_1} + C_2x^{\lambda_2} \qquad x \neq 0$ \vspace{2mm} \casehead{B. Single root} \vspace{1mm} \\ \soln: $y(x) = x^{\lambda}(C_1 + C_2\ln|x|)$ \vspace{2mm} \casehead{C. Complex roots} ($\lambda_{1,2} = \alpha \pm i\beta$) \vspace{1mm} \\ \soln: $y(x) = x^\alpha\left[C_1\cos(\beta \ln|x|) + C_2\sin(\beta \ln|x|)\right]$ \end{minipage} \end{tabular} } %--------------------------------------------------------------------------------------------------- \subsection*{2nd-order Non-Homogeneous} \begin{center} {$F(y'', y', y, x) = 0$ \qquad $y'' + a(x)y' + b(x)y = f(x)$ \qquad \soln: $y = y_h + y_p = C_1y_1(x) + C_2y_2(x) + y_p(x)$} \end{center} \interspace % 2nd-order linear methods -- simple methods \begin{tabular}{cc} \begin{minipage}[t]{0.5\textwidth} % Left column \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Simple case:\; $y', y$ missing} \begin{displaymath} y'' = f(x) \end{displaymath} \soln: Integrate twice. \end{minipage}} \interboxsep \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Simple case:\; $y', x$ missing} \begin{displaymath} y'' = f(y) \end{displaymath} \soln: Change of var: $p = y'$ + chain rule, then \\ $p\dfrac{dp}{dy} = f(y)$ is var.sep. \\ Solve it, back-replace $p$ and solve again. \end{minipage}} \end{minipage} \begin{minipage}[t]{0.5\textwidth} % Right column \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Simple case:\; $y$ missing} \begin{displaymath} y'' = f(y', x) \end{displaymath} \soln: Change of var: $p = y'$ and then solve twice. \end{minipage}} \interboxsep \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Simple case:\; $x$ missing} \begin{displaymath} y'' = f(y', y) \end{displaymath} \soln: Change of var: $p = y'$ + chain rule, then \\ $p\dfrac{dp}{dy} = f(p, y)$ is 1st-order ODE. \\ Solve it, back-replace $p$ and solve again. \end{minipage}} \end{minipage} \end{tabular} \interboxsep % 2nd-order linear methods -- advanced methods \begin{tabular}{cc} \begin{minipage}{0.5\textwidth} % Left column \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Method of Undetermined Coefficients / ``Guesswork''} \soln: Assume $y(x)$ has same form as $f(x)$ with \\ undetermined constant coefficients. \\ Valid forms: {\begin{enumerate}\tightitems \item $P_n(x)$ \item $P_n(x)e^{ax}$ \item $e^{ax}(P_n(x)\cos bx + Q_n(x) sin bx$ \end{enumerate}} \textit{Failure case:} If any term of $f(x)$ is a solution of $y_h$, \\ multiply $y_p$ by $x$ and repeat until it works. \end{minipage}} \end{minipage} \begin{minipage}{0.5\textwidth} % Right column \framebox{\begin{minipage}{\twocolwidth} \subsubsection*{Variation of Parameters (Lagrange Method)} (More general, but you need to know $y_h$) \\ \soln: $y_p = v_1y_1 + v_2y_2 + \cdots + v_ny_n$ \\ \vspace{1em} {\small $ \begin{array}{lclclcl} v'_1y_1 & + & \cdots & + & v'_ny_n & = & 0 \\ v'_2y'_2 & + & \cdots & + & v'_ny'_n & = & 0 \\ \cdots & + & \cdots & + & \cdots & = & 0 \\ v'_ny^{(n-1)}_b & + & \cdots & + & v'_ny^{(n-1)}_n & = & \phi(x) \end{array}$ \\ }\vspace{1em} Solve for all $v'_i$ and integrate. \end{minipage}} \end{minipage} \end{tabular} \clearpage %--------------------------------------------------------------------------------------------------- \subsection*{Power Series Solutions} \begin{tabular}{cc} \begin{minipage}[t]{0.5\textwidth} % Left column {\begin{enumerate}\tightitems \item Assume $y(x) = \sum_{n=0}^{\infty} c_n(x-a)^n$, compute y', y'' \item Replace in the original D.E. \item Isolate terms of equal powers \item Find \emph{recurrence relationship} between the coefs. \item Simplify using common series expansions \end{enumerate}} (Use $y = vx$, $z = v'$ to find $y_2(x)$ if only $y_1(x)$ is known.) \end{minipage} \hspace{0.03\textwidth} \begin{minipage}[t]{0.47\textwidth} % Right column \subsubsection*{Taylor Series variant} {\begin{enumerate}\tightitems \item Differentiate both sides of the D.E. repeatedly \item Apply initial conditions \item Substitute into T.S.E. for $y(x)$ \end{enumerate}} \end{minipage} \end{tabular} \interspace \begin{tabular}{cc} \begin{minipage}[t]{0.5\textwidth} % Left column \subsubsection*{Validity} For $y'' + a(x)y' + b(x)y = 0$ \\ if $a(x)$ and $b(x)$ are analytic in $|x| < R$, \\ the power series also converges in $|x| < R$. \interboxsep \textit{Ordinary Point}: Power method success guaranteed.\\ \textit{Singular Point}: success \emph{not} guaranteed. \end{minipage} \begin{minipage}[t]{0.45\textwidth} % Right column \interboxsep \textit{Regular} singular point: \\ if $x a(x)$ and $x^2 b(x)$ have a \emph{convergent MacLaurin series} near point $x=0$. (Use translation if necessary.) \interboxsep \textit{Irregular} singular point: otherwise. \end{minipage} \end{tabular} \interspace \subsubsection*{Method of Frobenius for Regular Singular pt.} \begin{tabular}{cc} \begin{minipage}[t]{0.5\textwidth} % Left column \[ y(x) = x^r(c_0 + c_1 x + c_2 x^2 + \cdots ) = \sum_{n=0}^{\infty} c_n x^{r+n} \] \begin{center} Indicial eqn: \quad $r(r-1) + a_0 r + b_0 = 0$ \end{center} \casehead{Case I:} $r_1$ and $r_2$ differ but \emph{not by an integer} \begin{eqnarray*} y_1(x) & = |x|^{r_1} \left( \sum_{n=0}^{\infty} c_nx^n \right), & c_0 = 1 \\ y_2(x) & = |x|^{r_2} \left( \sum_{n=0}^{\infty} c_n^*x^n \right), & c_0^* = 1 \end{eqnarray*} \end{minipage} \begin{minipage}[t]{0.45\textwidth} % Right column \casehead{Case II:} $r_1 = r_2$ \begin{eqnarray*} y_1(x) & = |x|^{r} \left( \sum_{n=0}^{\infty} c_nx^n \right), & c_0 = 1 \\ y_2(x) & = |x|^{r} \left( \sum_{n=1}^{\infty} c_n^*x^n \right) + y_1(x)ln|x| & \end{eqnarray*} \casehead{Case III:} $r_1$ and $r_2$ differ by an integer \begin{eqnarray*} y_1(x) & = |x|^{r_1} \left( \sum_{n=0}^{\infty} c_nx^n \right), & c_0 = 1 \\ y_2(x) & = |x|^{r_2} \left( \sum_{n=0}^{\infty} c_n^*x^n \right) + c_1^* y_1(x) ln|x|, & c_0^* = 1 \end{eqnarray*} \end{minipage} \end{tabular} \subsection*{Gamma Function} \begin{tabular}{cc} \begin{minipage}[t]{0.5\textwidth} % Left column \Gamma(x) = \end{minipage} \begin{minipage}[t]{0.45\textwidth} % Right column \end{minipage} \end{tabular} \subsection*{Laplace Transform} \TODO \subsection*{Fourier Transform} \TODO %------------------------------------------------------------------------------------------------------------- % Copyright/distribution license. {\vfill\hfill{\tiny Author: Martin Blais, 2009. This work is licensed under the Creative Commons ``Attribution - Non-Commercial - Share-Alike'' license.}} \end{document} %------------------------------------------------------------------------------------------------------------- %------------------------------------------------------------------------------------------------------------- %------------------------------------------------------------------------------------------------------------- % % Notes % % general Wronskian %$ W(y_1, y_2, \dots, y_n) = \left| \begin{array}{cccc} %y_1 & y_2 & \dots & y_n \\ %y'_1 & y'_2 & \dots & y'_n \\ %y''_1 & y''_2 & \dots & y''_n \\ %\vdots & \vdots & & \vdots \\ %y^{n-1}_1 & y^{n-1}_2 & \dots & y^{n-1}_n \end{array} \right|$ %\name{Abel's Formula} \\ %$W(y_1, y_2)(x) = Ce^{-\int a(x) dx}$ %-------------------------------------------------------------------------------------------------------------------------- % Summary of Techniques -- Eventually, all of these should be included on the cheat sheet. % % - Add a table of common integrating factors? Laplace transforms? % % - Add a section for numerical techniques % % - Annihilator methods --> near Variation of parameters % - D-Operator Method % - Similarity reduction method (from heat equation) % http://tutorial.math.lamar.edu/classes/de/de.aspx http://www.sosmath.com/diffeq/diffeq.html % % First Order Differential Equations First Order Differential Equations % % - Linear Equations - Linear Equations % - Separable Equations - Separable Equations % - Exact Equations - Qualitative Technique: Slope Fields % - Bernoulli Differential Equations - Equilibria and the Phase Line % - Substitutions - Bifurcations % - Intervals of Validity - Bernoulli Equations % - Modeling with First Order Differential Equations - Riccati Equations % - Equilibrium Solutions - Homogeneous Equations % - Euler’s Method - Exact and Non-Exact Equations % - Integrating Factor technique % - Some Applications % o Radioactive Decay % o Newton's Law of Cooling % o Orthogonal Trajectories % o Population Dynamics % - Numerical Technique: Euler's Method % - Existence and Uniqueness of Solutions % - Picard Iterative Process % % % Second Order Differential Equations Second Order Differential Equations % % - Basic Concepts - Nonlinear Equations % - Real Roots - Linear Equations % - Complex Roots - Homogeneous Linear Equations % - Repeated Roots - Linear Independence and the Wronskian % - Reduction of Order - Reduction of Order % - Fundamental Sets of Solutions - Homogeneous Equations with Constant Coefficients % - More on the Wronskian - Non-Homogeneous Linear Equations % - Nonhomogeneous Differential Equations - Method of Undetermined Coefficients % - Undetermined Coefficients - Method of Variation of Parameters % - Variation of Parameters - Euler-Cauchy Equations % - Mechanical Vibrations - Series Solutions % - Introduction % - Derivatives and Index Shifting % - First Examples % - Airy's Equation % - The Radius of Convergence of Series Solutions % - Hermite's Equation % % % Laplace Transforms Laplace Transform % % - The Definition - Basic Definitions and Results % - Laplace Transforms - Application to Differential Equations % - Inverse Laplace Transforms - Impulse Functions: Dirac Function % - Step Functions - Convolution Product % - Solving IVP’s with Laplace Transforms - Table of Laplace Transforms % - Nonconstant Coefficient IVP’s % - IVP’s with Step Functions % - Dirac Delta Function % - Convolution Integral % - Table of Laplace Transforms % % % Systems of Differential Equations Systems of Differential Equations % % - Review : Systems of Equations - Introduction and Motivation % - Review : Matrices and Vectors - Second Order Equations and Systems % - Review : Eigenvalues and Eigenvectors - Euler's Method for Systems % - Systems of Differential Equations - Qualitative Analysis % - Solutions to Systems - Linear Systems % - Phase Plane - Introduction and First Definitions % - Real Eigenvalues - Vector Representations of Solutions of Linear Systems % - Complex Eigenvalues - Eigenvalues and Eigenvectors Technique % - Repeated Eigenvalues + Real Eigenvalues % - Nonhomogeneous Systems + Repeated Eigenvalues % - Laplace Transforms + Complex Eigenvalues % - Modeling + Zero Eigenvalues % - Qualitative Analysis of Linear Systems % - Nonlinear Systems % - Equilibrium Point Analysis: Linearization Technique % % % Series Solutions % % - Review : Power Series % - Review : Taylor Series % - Series Solutions % - Euler Equations % % % Higher Order Differential Equations Higher Order Linear Equations % % - Basic Concepts for nth Order Linear Equations - Introduction and Basic Results % - Linear Homogeneous Differential Equations - Homogeneous Linear Equations with Constant Coefficients % - Undetermined Coefficients - Non-Homogeneous Linear Equations % - Variation of Parameters - Method of Undetermined Coefficients % - Laplace Transforms - Method of Variation of Parameters % - Systems of Differential Equations % - Series Solutions % % % Boundary Value Problems & Fourier Series Fourier Series % % - Boundary Value Problems - Fourier Series: Basic Results % - Eigenvalues and Eigenfunctions - Fourier Sine and Cosine Series % - Periodic Functions and Orthogonal Functions - Convergence of Fourier Series % - Fourier Sine Series - More on Convergence of Fourier Series % - Fourier Cosine Series - Gibbs Phenomenon % - Fourier Series - Bessel's Inequality and Parseval Formula: The Energy Theorem % - Convergence of Fourier Series - Operations on Fourier Series % - Application of Fourier Series to Differential Equations % % % Partial Differential Equations % % - The Heat Equation % - The Wave Equation % - Terminology % - Separation of Variables % % Solving the Heat Equation % % - Heat Equation with Non-Zero Temperature Boundaries % - Laplace’s Equation % - Vibrating String % - Summary of Separation of Variables % % Riccati equation