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{\bf MS COMPREHENSIVE EXAM \\ DIFFERENTIAL EQUATIONS \\ SPRING 2012} \\
Ivie Stein, Jr. and H. Westcott Vayo
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\vspace{.3in}

\noindent{\it This exam has two parts, (A) ordinary differential
equations and (B) partial differential equations. Do any three of the four problems in each part. Clearly indicate which three problems in each part are to be graded. Show the details of your work}.

\vspace{.8in}

\noindent{\bf Part A:  Ordinary Differential Equations}

\vspace{.2in}
\begin{enumerate}
%1
\item Consider the second order linear homogeneous ordinary differential equation
\[(1-x^2)y''-xy'+4y=0.\]

Assume a solution of the form
\[y=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+\cdots\]

to find two linearly independent solutions.

\vspace{.5in}
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%2
\item Let $y_1$ and $y_2$ be two differentiable functions with second derivatives defined on $(a,b)$. Show that the Wronskian \[W(y_1,y_2)(t)=\left|\begin{array}{ll}y_1(t) & y_2(t)\\ y'_1(t) & y'_2(t) \end{array}\right|\] is nonzero where $t$ is in $(a,b)$ if    $y_1$ and $y_2$ are linearly independent on interval $(a,b)$ and if $y_1$ and $y_2$ are solutions to $y''+p(t)y'+q(t)y=0$ where $p$ and $q$ are continuous on $(a,b)$.
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%3
\item Let $A=\left(\begin{matrix} 1 & 1\\ 9 & 1 \end{matrix}\right)$.

\begin{enumerate}
%a
\item Find all eigenvalues of $A$.\\

%b
\item For each eigenvalue of $A$, find all corresponding eigenvectors.\\

%c
\item Find the general solution to $x'=Ax$.

\end{enumerate}

\vspace{.5in}

%4
\item
\begin{enumerate}
\item Find all critical points of \[\begin{cases}
x'=-x+y-x(y-x)\\
y'=-x-y+2x^2y
\end{cases}\hspace{-.1in}.\]
\vspace{.3in}
%b
\item Classify the critical point $x=0$, $y=0$ as to the type and stability. Refer to the attached table 9.3.1. Provide a phase plane portrait.
\end{enumerate}

\end{enumerate}





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\noindent{\large\it You may work completely any three of the four problems.}

\vspace{.7in}

\noindent{\bf Part B:  Partial Differential Equations}\\

\vspace{.1in}

\begin{enumerate}

%1
\item Solve explicitly for $u(x,y)$:
\[u(xu_x-yu_y)=y^2-x^2.\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{.4in}

%2
\item
Show that $z(x,y)=f[u(x,y)]$ satisfies the equation
$a(x,y)z_x+b(x,y)z_y=0.$
%%%%%%%%%%%%%%%%%%%%%%%
\vspace{.4in}

%3
\item
The Cauchy-Riemann equations \[\begin{cases}
\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}=0\\
\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=0
\end{cases}\]
are a system of linear PDE's.  Write the system in matrix form and obtain the characteristic equation.
\vspace{.3in}






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%4
\item Given the \underline{two} initial-value problems
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
u_{xx} &- c^{-2}u_{tt} = 0 & -\infty &<x<\infty\notag\\
u(x,0) &= f(x) & 0 &<t <\infty\notag\\
u_t(x,0) &= 0 \notag
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%
and
\begin{align}
v_{xx} &- c^{-2}v_{tt} = 0 & -\infty &<x<\infty\notag\\
v(x,0) &= 0 & 0 &<t <\infty\notag\\
v_t(x,0) &= g(x). \notag
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $w(x,t)=u(x,t)+v(x,t)$ and formulate a new IVP in $w(x,t)$.






\end{enumerate}
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