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% Document Info
\author{stoneoxmike}
\pdfinfo{
  /Title (differential-equations.pdf)
  /Creator (Cheatography)
  /Author (stoneoxmike)
  /Subject (Differential Equations Cheat Sheet)
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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\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/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}stoneoxmike \\
  \uline{cheatography.com/stoneoxmike} \\
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  \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}.
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  Measure your website readability!\\
  www.readability-score.com
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\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Useful Formulas}}  \tn
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
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\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 
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\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Separable Linear Equations}}  \tn
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\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}}-}
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\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 
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\begin{tabularx}{8.4cm}{X}
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\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}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Exact Equations}}  \tn
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\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 
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\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 
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\par\addvspace{1.3em}


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

\end{document}