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% Cheatsheet for Differential Equations
%

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\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
