\documentclass[10pt,a4paper]{article}

% Packages
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\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
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\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{katalyst}
\pdfinfo{
  /Title (ode-not-being-updated.pdf)
  /Creator (Cheatography)
  /Author (katalyst)
  /Subject (ODE not being updated Cheat Sheet)
}

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% 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
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\newcommand{\tn}{\tabularnewline} % Required as custom column type in use

% Font and Colours
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\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{ODE not being updated Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{katalyst} via \textcolor{DarkBackground}{\uline{cheatography.com/162879/cs/34104/}}}
\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}katalyst \\
  \uline{cheatography.com/katalyst} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 14th October, 2022.\\
   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*}{3}

\begin{tabularx}{5.377cm}{p{0.64078 cm} x{1.51041 cm} x{2.42581 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{First order Differential Equations}}  \tn
% Row 0
\SetRowColor{LightBackground}
\seqsplit{Linear} & a(t)y' + b(t)y = f(t) & Normal form y' + p(t)y = q(t) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\seqsplit{Separable} & dy/dt = g(y)*h(t) &  \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\seqsplit{Bernoulli} & a(t)y' + b(t)y = f(t)y\textasciicircum{}m\textasciicircum{} & m≠ 0,1 \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
\seqsplit{Homogeneous} & y' = g(y/t) &  \tn 
% Row Count 9 (+ 3)
% Row 4
\SetRowColor{LightBackground}
Exact & M(x,y)dx + N(x,y)dy = 0 & Exact if and only if the partials My and Nx are equal \tn 
% Row Count 12 (+ 3)
% Row 5
\SetRowColor{white}
\seqsplit{Non-Exact} & M(x,y)dx + N(x,y)dy = 0 & When My≠Nx \tn 
% Row Count 14 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{5.377cm}}{First order DE's and their form}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{3.28482 cm} x{1.69218 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Solving first order linear}}  \tn
% Row 0
\SetRowColor{LightBackground}
1. Make sure its in normal form & y' + p(t)y = q(t) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
2. Find an integrating factor & µ(t) = e\textasciicircum{}∫p(t)dt\textasciicircum{} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
3. Multiply both sides of the normal form by µ(t) to get & (µ(t)y)' = µ(t)q(t) \tn 
% Row Count 7 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{4. Integrate both sides of (µ(t)y)' = µ(t)q(t) and solve for y} \tn 
% Row Count 9 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Dont Forget constants of integration}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Solving FO Separable DE}}  \tn
% Row 0
\SetRowColor{LightBackground}
1. Rewrite y' and dy/dt and separate the variable y from the variable t to get d & dy/dt =g(y)h(t) ~ where we get... \{\{nl\}\} (1/g(y)) dy = h(t)dt \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{2. Integrate both sides to obtain} \tn 
% Row Count 5 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.00694 cm} x{1.55618 cm} x{2.01388 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Second and HIgher Order DE's}}  \tn
% Row 0
\SetRowColor{LightBackground}
2\textasciicircum{}ND\textasciicircum{} Order Linear & a(t)y'' + b(t)y' + c(t)y = f(t) & Normal form y'' + p(t)y' + q(t)y = r(t) \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\seqsplit{Homogeneous} (H) & a(t)y'' + b(t)y' + c(t)y = 0 & Gen. Soltn. yH(t,c1,c2) \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\seqsplit{Non-Homogeneous} (NH) & a(t)y'' + b(t)y' + c(t)y = f(t) & Gen. Soltn. yH(t,c1,c2) + yp(t) \tn 
% Row Count 9 (+ 3)
% Row 3
\SetRowColor{white}
(H) const. coeff. & ay'' + by' + cy = 0 & a≠0, b,c are consts. \tn 
% Row Count 12 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\seqsplit{Cauchy-Euler} & at\textasciicircum{}2\textasciicircum{}y'' + bty' + cy = 0 & a≠0, b,c are consts. \tn 
% Row Count 14 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{5.377cm}}{yH(t) = general solution of (H) \newline yP(t) = particular solution of (NH)}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}