\documentstyle[12pt]{article}
\pagestyle{empty}
\parindent=0 in
\begin{document}
\begin{center}
{\bf \large Ph.D. Qualifying EXAM \\ DIFFERENTIAL EQUATIONS \\ SPRING 2001}
\\Biao Ou and Westcott Vayo
\end{center}
This exam has two parts, ordinary differential equations and partial differential equations. Do all the problems.
\vspace{24 pt}
{\bf Part I: Ordinary Differential Equations}
\vspace{12 pt}
\newline
{\bf 1.} Given the non-autonomous system as below.
Show that the paths followed by particles emitted at $(x_{0},y_{0}) $ at
$t=s$ are the same regardless of the value of $s.$ Why is this so?
\[ \frac{dx}{dt} = \frac{x}{1+t},\;\;\;\;. \frac{dy}{dt} = \frac{y}{1+t}.
\]
\newline
{\bf 2.} Consider the system
\[ \dot{x} = x, \;\;\;\; \dot{y} = -y + x^{2}.
\]
(a) Determine the stability and type of the rest point (0,0).
(b) Find a function $\phi(x,y)$ such that all solutions are level
curves $\phi(x,y)=c.$
\vspace{12 pt}
\newline
{\bf 3.} Let $y_{1}(x), y_{2}(x)$ be two linearly independent solutions of
the equation $ y'' + p(x)y' + q(x)y = 0, $ where $p(x),q(x)$ are continuous
functions on $(-\infty,\infty).$ Prove that all the roots of $y_{1}(x),y_{2}(x)$ are simple and that between every two roots of $y_{1}(x)$ there is
exactly one root of $y_{2}(x).$
\newpage
{ \bf Part II: Partial Differential Equations}
\vspace{12 pt}
\newline
{\bf 1.} Given the initial value problem
\begin{eqnarray*}
u_{tt} & = & c^{2} u_{xx} \;\;\;\mbox{on}\;\;\; t>0; \\
u(x,0) & = & f(x), \\
u_{t}(x,0) & = & g(x);
\end{eqnarray*}
make the change of variables $ \xi=x+ct, \;\zeta=x-ct $ and solve the
equation and its initial conditions.
\vspace{12 pt}
\newline
{\bf 2.} Given the second order PDE
\[ U_{xx}-3U_{yy}+2U_{x}-U_{y}+U=0; \;\;U=U(x,y)
\]
identify the type of equation and transform it into a canonical form consistent with its type as identified above.
\vspace{12 pt}
\newline
{\bf 3.} Consider the Laplace equation $ \Delta u = u_{xx}+u_{yy} $ on the
square $ 0 \leq x \leq \pi, \;0\leq y \leq \pi $ with the boundary conditions
\[ u(0,y)=u(\pi,y)=0,\;\;u(x,0)=0,\;\;u(x,\pi)=f(x).
\]
Use the method of separation of variables to find a series solution and decide the coefficients of the series.
\end{document}