
\documentclass[12pt]{amsart}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{float}
\usepackage{amsmath}
\usepackage{graphicx}

\setcounter{MaxMatrixCols}{10}
%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=5.50.0.2960}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{BibliographyScheme=Manual}
%TCIDATA{LastRevised=Monday, February 22, 2021 08:53:10}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{CSTFile=amsart.cst}

\input{tcilatex}
\evensidemargin=0in
\oddsidemargin=0in
\topmargin=-.70in
\setlength{\parskip}{.05in}
\setlength{\parindent}{0in}
\textwidth=6.45in
\textheight=9.0in

\begin{document}

\begin{center}
\textbf{Math 2233.21570}\\[0pt]
FIRST EXAM \\[0pt]
February 23, 2021
\end{center}

\begin{flushright}
Name:\underline{\hspace{0.5in}Kale Downing \hspace{0.5in}}
\end{flushright}

1. (15 pts) Classify the following differential equations: determine their order, if
they are linear or non-linear, and if they are ordinary differential
equations or partial differential equations.

(a) $\displaystyle\frac{dx^{2}}{dt^{2}}+tx=\sin \left( x\right) $

\vspace{.2in}

(b) $\displaystyle\frac{\partial ^{3}\psi }{\partial ^{3}x}-x^{2}\frac{%
\partial \psi }{\partial y}=\psi ^{2}$

\vspace{.2in}

(c) $\displaystyle\frac{d^{3}x}{dt^{3}}+t^{2}\frac{dx}{dt}+x=0$

\vspace{.2in}

(d) $\displaystyle x^{2}y^{\prime }+y=e^{x}\sin \left( x\right) $

\vspace{.2in}

(e) $\displaystyle\frac{\partial ^{2}\phi }{\partial x^{2}}+\frac{\partial
\phi }{\partial y}=\left( x+y\right) ^{2}$

\vspace{.2in}

2. (15 pts) Consider the following first order ODE: $y^{\prime }=xy$ and suppose $%
y(x)$ is the solution satisfying $y(1)=2.$ Use the numerical (Euler) method
with $n=3$ and $\Delta x=0.1$ to estimate $y(1.3)$.

\newpage

3. (15 pts) Find an explicit solution of the following (separable) differential
equation. 
\begin{equation*}
3x^{2}-e^{2y}y^{\prime }=x
\end{equation*}

\vspace{3.5in}

4. (15 pts) Solve the following initial value problem 
\begin{equation*}
xy^{\prime }-2y=x^{2}\qquad ,\qquad y(1)=2
\end{equation*}

\newpage

5. (20 pts) Consider the following initial value problem%
\begin{eqnarray*}
3x^{2}+\left( 2y-2x\right) \frac{dy}{dx} &=&2y \\
y\left( 1\right) &=&2
\end{eqnarray*}%
Show that the differential equation is exact and then find the explicit
solution satisfying the initial condition.

\vspace{3.5in}

6. (20 pts) Find the general solution of%
\begin{equation*}
y\frac{dy}{dx}=x+y
\end{equation*}%
(Hint: recast the differential equation into the form $\frac{dy}{dx}=F\left( 
\frac{y}{x}\right) $ and then use the change of variable $z\left( x\right) =%
\frac{y\left( x\right) }{x}$.)

\end{document}
