% S2019, 2280-1
\documentclass[11pt]{article}

%\documentclass[]{article}
%\usepackage{array}
%\usepackage{amsmath}
%\usepackage{amsfonts}
\usepackage{amssymb}
%\usepackage{amsxtra}
%\usepackage{epsfig}
%\setlength{\topmargin}{-.3 in}
%\setlength{\oddsidemargin}{0in}
%\setlength{\evensidemargin}{0in}
%\setlength{\textheight}{9.5in}
%\setlength{\textwidth}{6.5in}


 \setlength{\textheight}{10.4in}
 \setlength{\topmargin}{-1in}
 \setlength{\headheight}{0in}
 \setlength{\topsep}{0in}
 \setlength{\marginparwidth}{0in}
 \setlength{\marginparsep}{0in}
% \setlength{\footskip}{4pc}
%\setlength{\footheight}{0in}
 \setlength{\oddsidemargin}{0in}
 \setlength{\evensidemargin}{0in}
 \setlength{\textwidth}{6.4in}
 \setlength{\parskip}{4pt}
 \setlength{\parindent}{0pt}
\def\MAPLE{{\tt Maple}}
\def\MATLAB{{\tt Matlab}}
\def\MAXIMA{{\tt Maxima}}
\def\MATHEMATICA{{\tt Mathematica}}
\def\dd{\displaystyle}

% \begin{topics}{template} ... \end{topics}
% used like enumerate, itemize, description environments
% The item[] is never optional!
%
 \newcommand{\topicslabel}[1]{\mbox{#1}\hfil}
 \newenvironment{topics}[1]{
    \begin{list}{}{%
      \let\makelabel\topicslabel
      \settowidth{\labelwidth}{\topicslabel{#1}}
      \setlength{\leftmargin}{\labelwidth}
      \advance\leftmargin by 1.5em}
}{\end{list}}
\usepackage{alltt}
\usepackage{multicol}
%\usepackage{email}
\usepackage{hyperref}

\begin{document}
\begin{center} \large\sf Week 2 Examples\end{center}

{\bf Example 1}: Classify as separable (S), quadrature (Q), linear (L) or none (N). (1) $y'=3(xy)^{1/3}$, (2) $y'=xy^2+1$, (3) $y'=x\sin(y)$, (4) $y'=y\sin(x)$, (5) $y'=e^{\ln|x|}$, (6) $y'+xy=x^2y$\newline
{\bf Answers}: (1) S; (2) N; (3) S; (4) S,L; (5) Q,S,L; (6) L.

{\bf Example 2}: Check {\bf explicit} answer $y=(x^{3/2}+c)^2$ for $y'=3\sqrt{x}\sqrt{y}$ on domain $x\ge 0$, $y\ge 0$.

{\bf Example 3}: Check {\bf implicit} answer $\csc(y)\cot(y)=-x^2/2+c$ for $y'=x\sin(y)$.

{\bf Example 4}: Let $f(x,y)=1-x^2+y^2-x^2y^2$. In relation $f(x,y)=F(x)G(y)$, equations $f(x,0)=F(x)G(0), f(0,y)=F(0)G(y)$ can determine $F,G$ . Explain. Then find one pair $F,G$.

{\bf Example 5}: Solve using the constant equation shortcut or the  quadrature shortcut.\newline
(1) $y'+2y=6$, (2)  $2y'+5y=3$, (3) $2y'=3$, (4) $3y'=5y + \pi$.

{\bf Example 6}: Solve using the integrating factor shortcut for homogeneous equations.\newline
 (1) $y'+8xy=0$,  (2) $2y'+\sin(x)y=0$, (3) $xy'+\ln|x| y=0$.

{\bf Example 7}: Solve a non-separable equation using the integrating factor method.\newline (1) $xy'+2y=x^2$ , (2) $xy'+2y=x$, (3) $xy'+2y\ln|x| = \ln|x|\,e^{(\ln|x|)^2}$.\newline
{\bf Answers}: (1) $y=x^2/4+c/x^2$, (2) $y=x/3 + c/x^2$, (3) $y=\frac14\,e^{(\ln|x|)^2} + c/e^{(\ln|x|)^2}$.
% de:=diff(y(x),x)*x+2*y(x) =x^2;dsolve(de,y(x));
% de:=diff(y(x),x)*x+2*y(x) =x;dsolve(de,y(x));
% de:=diff(y(x),x)*x+2*ln(x)*y(x) = ln(x)*exp((1)*(ln(x))^2);
% dsolve(de,y(x));

{\bf Example 8}: Solve the brine tank model $\frac{dx}{dt}=1/4-x/16$, $x(0)=20$.

{\bf Example 9}: Solve the brine tank cascade $x'=-x/2$, $y'=x/2-y/4$, $z'=y/4-z/6$ with $x(0)=1,y(0)=-2, z(0)=1.5$. {\bf Answer}: $x=e^{-t/2}$, $y=-2e^{-t/2}$, $z=1.5e^{-t/2}$

{\bf Example 10}: Find all equilibrium solutions for $(x^2+1)y'=x+1-xy^2-y^2$

{\bf Example 11}: Solve $y'=(1-y)y$ by the substitution $u=y/(1-y)$.

{\bf Example 12}: Solve $y'=(1-y)y$ by partial fraction methods.

{\bf Example 13}: Solve $y'=7y(y-13)$, $y(0)=17$. See 2.1-8.

{\bf Example 14}: Draw a phase line diagram for $y'=y(1-y)^2(y+1)$.

{\bf Example 15}: Draw a phase diagram for $y'=y^2(y^2-4)$. See 2.2-17.


\end{document}
