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{\Large\bf Differential Equations - M.S. Exam} \\
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Spring 2003 \hspace{2.5in} B. Ou and W. Vayo
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The Examination Committee tries to proofread the exams as carefully as possible. Nevertheless, the exam may contain misprints. If you are convinced a problem has been stated incorrectly, mention this to the proctor and indicate your interpretation in your solution. In such cases do not interpret the problem in such a way that it becomes trivial.
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{\bf\underline{M.S. Comprehensive Examination}}
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{\slshape\noindent You should work \underline{any three} of the four problems on each of the two parts (ODE, PDE). Show all your work and clearly indicate your answers.}
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\noindent\underline{PART ODE}
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%1
\item
Find three linearly independent solutions of the equation
\[y'''(t)+y'(t)-2y(t)=0\]
\noindent and prove that they are linearly independent.
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\item Show that there are no negative eigenvalues for \mbox{$\phi''(x)=- \lambda \phi(x)$,} where $\phi'(0)=0$ and $\phi'(L)=0$ on $[0, \, L]$.
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\item Consider the system
\[\frac{dx}{dt}=ax+by, \; \frac{dy}{dt}=cx+dy \; (a,b,c,d \mbox{ real}).\]
Show that if $ad-bc \neq 0$, the only equilibrium point is $(0, \, 0)$.
Show that if $ad-bc=0$, there are an infinite number of equilibrium points. Is $(0, \, 0)$ an ``isolated" equilibrium point? Why?
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\item Locate the critical points for the nonlinear system
\begin{equation}
\begin{array}{rcl} x'(t) & = & x(1-y)\\
y'(t) & = & y(1-2x). \notag \end{array}
\end{equation}
Determine the stability of the system for each of its critical points and sketch the trajectories, all on the same set of axes.
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\noindent\underline{PART PDE}
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\item Solve the first-order equation with the condition:
\[u_x=4u_y, \, u(0, \, y)=8e^{-3y} .\]
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\item Using the idea of superposition find two \underline {subproblems} that are easier to solve than is
\[\begin{array}{lc} u_t=u_{xx}+ \sin \pi x & 0