\documentclass[10pt,a4paper]{article} % Packages \usepackage{fancyhdr} % For header and footer \usepackage{multicol} % Allows multicols in tables \usepackage{tabularx} % Intelligent column widths \usepackage{tabulary} % Used in header and footer \usepackage{hhline} % Border under tables \usepackage{graphicx} % For images \usepackage{xcolor} % For hex colours %\usepackage[utf8x]{inputenc} % For unicode character support \usepackage[T1]{fontenc} % Without this we get weird character replacements \usepackage{colortbl} % For coloured tables \usepackage{setspace} % For line height \usepackage{lastpage} % Needed for total page number \usepackage{seqsplit} % Splits long words. %\usepackage{opensans} % Can't make this work so far. Shame. Would be lovely. \usepackage[normalem]{ulem} % For underlining links % Most of the following are not required for the majority % of cheat sheets but are needed for some symbol support. \usepackage{amsmath} % Symbols \usepackage{MnSymbol} % Symbols \usepackage{wasysym} % Symbols %\usepackage[english,german,french,spanish,italian]{babel} % Languages % Document Info \author{stoneoxmike} \pdfinfo{ /Title (differential-equations.pdf) /Creator (Cheatography) /Author (stoneoxmike) /Subject (Differential Equations Cheat Sheet) } % Lengths and widths \addtolength{\textwidth}{6cm} \addtolength{\textheight}{-1cm} \addtolength{\hoffset}{-3cm} \addtolength{\voffset}{-2cm} \setlength{\tabcolsep}{0.2cm} % Space between columns \setlength{\headsep}{-12pt} % Reduce space between header and content \setlength{\headheight}{85pt} % If less, LaTeX automatically increases it \renewcommand{\footrulewidth}{0pt} % Remove footer line \renewcommand{\headrulewidth}{0pt} % Remove header line \renewcommand{\seqinsert}{\ifmmode\allowbreak\else\-\fi} % Hyphens in seqsplit % This two commands together give roughly % the right line height in the tables \renewcommand{\arraystretch}{1.3} \onehalfspacing % Commands \newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour \newcommand{\mymulticolumn}[3]{\multicolumn{#1}{>{\columncolor{\RowColorName}}#2}{#3}} % For coloured multi-cols \newcolumntype{x}[1]{>{\raggedright}p{#1}} % New column types for ragged-right paragraph columns \newcommand{\tn}{\tabularnewline} % Required as custom column type in use % Font and Colours \definecolor{HeadBackground}{HTML}{333333} \definecolor{FootBackground}{HTML}{666666} \definecolor{TextColor}{HTML}{333333} \definecolor{DarkBackground}{HTML}{A3A3A3} \definecolor{LightBackground}{HTML}{F3F3F3} \renewcommand{\familydefault}{\sfdefault} \color{TextColor} % Header and Footer \pagestyle{fancy} \fancyhead{} % Set header to blank \fancyfoot{} % Set footer to blank \fancyhead[L]{ \noindent \begin{multicols}{3} \begin{tabulary}{5.8cm}{C} \SetRowColor{DarkBackground} \vspace{-7pt} {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}} } \end{tabulary} \columnbreak \begin{tabulary}{11cm}{L} \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Differential Equations Cheat Sheet}}}} \\ \normalsize{by \textcolor{DarkBackground}{stoneoxmike} via \textcolor{DarkBackground}{\uline{cheatography.com/138634/cs/29229/}}} \end{tabulary} \end{multicols}} \fancyfoot[L]{ \footnotesize \noindent \begin{multicols}{3} \begin{tabulary}{5.8cm}{LL} \SetRowColor{FootBackground} \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}} \\ \vspace{-2pt}stoneoxmike \\ \uline{cheatography.com/stoneoxmike} \\ \end{tabulary} \vfill \columnbreak \begin{tabulary}{5.8cm}{L} \SetRowColor{FootBackground} \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}} \\ \vspace{-2pt}Published 21st September, 2021.\\ Updated 21st September, 2021.\\ Page {\thepage} of \pageref{LastPage}. \end{tabulary} \vfill \columnbreak \begin{tabulary}{5.8cm}{L} \SetRowColor{FootBackground} \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}} \\ \SetRowColor{white} \vspace{-5pt} %\includegraphics[width=48px,height=48px]{dave.jpeg} Measure your website readability!\\ www.readability-score.com \end{tabulary} \end{multicols}} \begin{document} \raggedright \raggedcolumns % Set font size to small. Switch to any value % from this page to resize cheat sheet text: % www.emerson.emory.edu/services/latex/latex_169.html \footnotesize % Small font. \begin{multicols*}{2} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Useful Formulas}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/stoneoxmike_1632255884_Screenshot 2021-09-21 162422.png}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Recognize the Type}} \tn \SetRowColor{white} \mymulticolumn{1}{x{8.4cm}}{1. First check to see if it is separable. \newline % Row Count 1 (+ 1) 2. Next check to see if it is linear by looking at what is being done to the dependent variable and its derivatives. \newline % Row Count 4 (+ 3) 3. If it is not linear, but looks close, try Bernoulli form. \newline % Row Count 6 (+ 2) 4. Next, try checking for exactness. \newline % Row Count 7 (+ 1) 5. If x and y are only to the first power with the same coefficients, try substitution. \newline % Row Count 9 (+ 2) 6. If none of these work, try checking if it is separable again.% Row Count 11 (+ 2) } \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Separable Linear Equations}} \tn \SetRowColor{white} \mymulticolumn{1}{x{8.4cm}}{1. Check that the dependent variable (the one having its derivative taken) is only to the first power. \newline % Row Count 3 (+ 3) 2. Check that the dependent variable is not in a function (trig, exponential, log). \newline % Row Count 5 (+ 2) 3. Check that equation can be reorganized so that each variable is on opposite sides by itself. \newline % Row Count 7 (+ 2) 4. Integrate both sides and solve for dependent variable (don't forget C).% Row Count 9 (+ 2) } \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Non-separable Linear Equations}} \tn \SetRowColor{white} \mymulticolumn{1}{x{8.4cm}}{1. Reorganize into general form: \newline % Row Count 1 (+ 1) . \newline % Row Count 2 (+ 1) . \newline % Row Count 3 (+ 1) . \newline % Row Count 4 (+ 1) 2. Find the integrating factor: \newline % Row Count 5 (+ 1) . \newline % Row Count 6 (+ 1) . \newline % Row Count 7 (+ 1) . \newline % Row Count 8 (+ 1) 3. Use integrating factor in solved formula: \newline % Row Count 9 (+ 1) . \newline % Row Count 10 (+ 1) . \newline % Row Count 11 (+ 1) .% Row Count 12 (+ 1) } \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Bernoulli Equations}} \tn \SetRowColor{white} \mymulticolumn{1}{x{8.4cm}}{1. Reorganize equation into general form: \newline % Row Count 1 (+ 1) . \newline % Row Count 2 (+ 1) . \newline % Row Count 3 (+ 1) . \newline % Row Count 4 (+ 1) 2. Substitute v: \newline % Row Count 5 (+ 1) . \newline % Row Count 6 (+ 1) . \newline % Row Count 7 (+ 1) . \newline % Row Count 8 (+ 1) 3. Solve resulting linear equation: \newline % Row Count 9 (+ 1) . \newline % Row Count 10 (+ 1) . \newline % Row Count 11 (+ 1) . \newline % Row Count 12 (+ 1) 4. Solve for v and resubstitute.% Row Count 13 (+ 1) } \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Exact Equations}} \tn \SetRowColor{white} \mymulticolumn{1}{x{8.4cm}}{1. Reorganize into general form: \newline % Row Count 1 (+ 1) . \newline % Row Count 2 (+ 1) . \newline % Row Count 3 (+ 1) . \newline % Row Count 4 (+ 1) 2. Test for exactness with partial derivatives: \newline % Row Count 5 (+ 1) . \newline % Row Count 6 (+ 1) . \newline % Row Count 7 (+ 1) . \newline % Row Count 8 (+ 1) 3. Find f(x,y) for both M and N with partial integration: \newline % Row Count 10 (+ 2) . \newline % Row Count 11 (+ 1) . \newline % Row Count 12 (+ 1) . \newline % Row Count 13 (+ 1) 4. Find the general solution, including any terms that are missing from either integration: \newline % Row Count 15 (+ 2) . \newline % Row Count 16 (+ 1) . \newline % Row Count 17 (+ 1) .% Row Count 18 (+ 1) } \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Substitution Equations}} \tn \SetRowColor{white} \mymulticolumn{1}{x{8.4cm}}{1. Reorganize into general form: \newline % Row Count 1 (+ 1) . \newline % Row Count 2 (+ 1) . \newline % Row Count 3 (+ 1) . \newline % Row Count 4 (+ 1) 2. Let z equal: \newline % Row Count 5 (+ 1) . \newline % Row Count 6 (+ 1) . \newline % Row Count 7 (+ 1) . \newline % Row Count 8 (+ 1) 3. Find dz/dx: \newline % Row Count 9 (+ 1) . \newline % Row Count 10 (+ 1) . \newline % Row Count 11 (+ 1) . \newline % Row Count 12 (+ 1) 4. Substitute and solve resulting separable equation for dy/dx.% Row Count 14 (+ 2) } \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} % That's all folks \end{multicols*} \end{document}