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{\bf \large Ph.D. QUALIFYING EXAM \\ DIFFERENTIAL EQUATIONS \\ Spring, 2004}
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This exam has two parts, ordinary differential equations and partial
differential equations. In Part I, do problems 1 and 2 and choose two
from the remaining problems. In Part II, Choose three problems.
\vspace{24 pt}
{\bf Part I: Ordinary Differential Equations}
\vspace{12 pt}
\newline
{\bf 1.} Consider the differential equation with initial condition
\[ dx/dt = F(t,x),\;\; x(a)=x_{0} \in R^{n}
\]
where $x(t)=(x_{1}(t), x_{2}(t), \ldots, x_{n}(t))^{T} $ and
\newline
$F(t,x)=(F_{1}(t,x), F_{2}(t,x), \ldots, F_{n}(t,x))^{T}. $
Suppose $F(t,x) $ is continuous for $a \leq t \leq b$ and $ x \in R^{n}$ and \
satisfies a Lipschitz condition
$ |F(t,x)-F(t,y)| \leq L |x-y| $
for $ a \leq t \leq b $ and all $x,y.$
(a) Convert the differential equation with the initial condition into an equivalent integral equation.
(b) Set up the Picard iteration process and prove that the sequence converges
uniformly on the interval [a, b] to a limit function $x_{\infty}(t).$
(c) Show that $x_{\infty}(t) $ is a solution to the differential equation on
[a, b].
(d) Establish that the solution to the differential equation with the given initial condition is unique.
\vspace{12 pt}
\newline
{\bf 2.} \hspace{3 pt}(a) State and prove the Sturm Separation Theorem and Sturm Comparison
Theorem for O.D.E's of the form $y''+p(x)y=0.$
(b) Consider any non-trivial
solution to $y''+xy = 0.$ Show that $y(x)$ has infinitely many zeros on
$(0,\infty)$ but at most one zero on $(-\infty,0).$
\vspace{12 pt}
\newline
{\bf 3.} For the system show that (1,1) and (-1,-1) are its critical points.
Linearize the DE about each of the critical points and use the result to
describe the nature of the solutions in a neighborhood of the critical points.
\[
\dot{x} = x - y \;\;\;\;\;\; \dot{y} = 4 x^2 + 2 y^{2} - 6
\]
\newpage
{\bf 4.} Find all the eigenvalues and eigenfunctions of the Sturm-Liouville
system. Be sure to consider the possibility of $ \lambda \leq 0.$
\[ y''(t) + \lambda y(t) = 0;\;\;\;\; y(0)=0, \;\; y'(1)= 2 y(1).
\]
\newline
{\bf 5.} Find the solution $u(x,y)$ to the integrable system
on the domain $ \{ (x,y) \;|\; x<1, y< 1.\} $ with the initial
condition.
\[ u_{x} = \frac{u}{ 2(x+1) }, \;\; u_{y} = \frac{u}{y+1}; \;\;\;u(0,0)=1.
\]
\newline
{\bf 6.} Find the fundamental solution matrix for the
linear homogeneous system of differential equations. Sketch the possible
trajectories.
\[ \frac{d}{dt}
\left ( \begin{array}{c} x(t) \\ y(t) \end{array}
\right ) =
\left ( \begin{array}{cc}
3 & -2 \\
5 & -4
\end{array}
\right )
\left ( \begin{array}{c} x(t) \\ y(t) \end{array}
\right )
\]
\vspace{12 pt}
{\bf Part II: Partial Differential Equations}
\vspace{12 pt}
\newline
{\bf 1.} Solve the quasilinear partial differential equation by the method
of characteristics.
\[ u_{x} + u_{y} = u^{2}, \;\;\;\; u(x,0) = x^{2}.
\]
\newline
{\bf 2.} Consider the partial differential equation $u_{y}=u_{x}^{3}.$
(a) Find generalized solutions $u=G(x,y,a,b)$ which are linear in $x$
and $y.$
(b) Use the method of envelopes to solve the initial value problem
when $u(x,0)=x^{2}.$
\vspace{12 pt}
\newline
{\bf 3.} Derive (from scratch) the D'Alembert solution to the one-dimensional
wave equation
\begin{eqnarray*}
&& u_{tt}=c^{2}u_{xx} \;\;\mbox{for}\;\;x\in R \;\;\mbox{and}\;\;t>0 ;\\
&& u(x,0)=f(x),\;\;u_{t}(x,0)=g(x)
\end{eqnarray*}
where $f(x),g(x) \in C^{2}(R).$
\vspace{12 pt}
\newline
{\bf 4.} Suppose $ u(x) $ is a harmonic function on an open domain $\Omega.$
Prove that if $u(x)$ has a maximum at an interior point then $u(x)$
is a constant function.
\vspace{12 pt}
\newline
{\bf 5.} Let $ u(x) $ be a harmonic function on an entire Euclidean
space $ R^{n}. $ Suppose there exist two positive constants $c_{1}, c_{0}$
such that
\[ |u(x)| \leq c_{1} |x| + c_{0}
\]
for all $x$ in $R^{n}.$ Prove that $u(x)$ is a linear function.
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