Suggestions for Mathematical Exposition in Video & Writing

It's important that students learn how to solve problems and clearly communicate the thought process behind their solutions to others, usually through handwritten or typeset mathematical exposition, maybe with a few figures here and there to highlight relevant geometric aspects of the solution. Since I teach entirely online (I'm never in the classroom with my students) I feel that precise communication of ideas and solution methods are particularly important. There are often many paths to the correct answer in AP Calc, Linear Algebra, Multivariable Calc and Diffeqs. I want to know what insights and methods my students used to arrive at their answers, correct or otherwise. 

I took a relatively simple problem from differential equations 

y' = (3x^2-e^x)/(2y-5), y(0) = 1

and created a video explaining how to solve it, without me writing out every single step as I went along because I wanted to include some quick highlights during the solving itself and follow the solution with some qualitative dynamic exploration of the slope field and initial conditions using various free software such as Geogebra, dfield, and winplot.

After that, I wrote up a LaTeX document (stored on http://www.sharelatex.com)  and made a video describing some basic style & exposition guidelines for writing up one's HW or Tests in LaTeX. The same guidelines apply if writing solutions by hand.

Below is the raw LaTeX code and here is the document in PDF form.

% STYLE SUGGESTIONS
%
% When showing steps in compuations, line up equal signs vertically, one per line.
% Indent your code to make it more readable (and editable).
% Show relevant (calculus & diffeq) steps. 
% Interpret solution and steps in context of the problem & class. 
% Communicate your thought process so the reader does not have to assume or guess at what you did. 
%


\documentclass{article}


% Packages
\usepackage{amsmath, amssymb}   % math formatting & symbols
\usepackage{graphicx}           % insert graphics
\usepackage{eulervm, bookman}   % fonts for math & symbols
%\usepackage{fullpage}          % fullpage margins


\begin{document}
% Title 
\title{Seperable Differential Equation with Initial Value Example}
\author{Joe DiNoto}
\maketitle


\begin{enumerate}
    \item % 1 
        Given differential equation $y' = \dfrac{3x^2-e^x}{2y-5}$ and initial value $y(0)=1$
        \begin{enumerate}
            \item % 1a
                Solve the IVP explicitly, ($y$ as a function of $x$).
                The equation is seperable. Separate the variable and integrate. 
                \begin{align*}
                    \frac{dy}{dx}   & =\frac{3x^2-e^x}{2y-5}\\
                    \int 2y -5 \ dy & = \int 3x^2 -e^x \ dx \\
                    y^2 -5y +c_1    & = x^3 -e^x +c_2 \\
                    y^2 -5y         & = x^3 -e^x +C 
                \end{align*}    
                Combine the constants $c_2-c_1=C$. Now use the intial conditions to solve for $C$.
                \begin{align*}
                    (1)^2-5(1)  & = (0)^3-e^{(0)}+C\\
                                & \therefore \\
                            C   & = -3
                \end{align*}
                We now want to find express $y$ as a function of $x$. We can do this by noting that we ultimately have a quadratic equation in $y$. Use the quadratic formula to solve it for $y$. 
                \begin{align*}
                    y^2-4y -(x^3-e^x-3)     & = 0 \\
                    y                       & = \frac{-(-5) \pm \sqrt{(-5)^2-4(1)(-(x^3-e^x-3))}}{2(1)}\\
                    y                       & = \frac{5}{2} \pm \sqrt{13/4 + x^3-e^x}
                \end{align*}
                Again we use the intial conditions to determine if we should use the positive, negative, or both parts of the square root. 
                \begin{align*}
                    1   & = 5/2 \pm \sqrt{13/4 + (0)^3 -e^{(0)}} \\
                    1   & = 5/2 \pm 3/2 \\
                        & \therefore \\
                    y   & = \frac{5}{2} - \sqrt{13/4 + x^3-e^x}
                \end{align*}
                Because $5/2-3/2 = 1$ and $5/2+3/2 = 4 \neq 1$.
            \item % 1b
                Clearly sketch and label the solution curve and direction field. 
                \begin{center}
                    \includegraphics[scale=0.5]{figure.png}
                \end{center}
            \item % 1c
                What is the interval of validity for the function $y$ given the initial value?
                The function is defined only when the follwing is true: 
                    $$13/4 + x^3-e^x>0$$ 
                Using a computer we find that $x \in (-1.44, 4.63)$.
        \end{enumerate}
    \item % 2 
        Second question goes here
        \begin{enumerate}
            \item % 2a
                two a goes here
            \item % 2b 
                two b goes here
        \end{enumerate}
    \item % 3
        Third question here
    \item % 4 
        Fourth question goes here
\end{enumerate}




\end{document}