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\centerline{Mathematics 304, section 2 -- Ordinary Differential Equations}
\centerline{Practice Final Exam}
\centerline{December 13, 2004}
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\noindent
I.  Consider the ODE 
$$y' = {y\over t^2}$$
\item{A)}  SHow that the constant function $y_1(t) = 0$ is a
solution.
\item{B)}  Show that there are infinitely many other functions
that satisfy this equation, that agree with $y_1(t)$ for
$t\le 0$, but that are nonzero when $t > 0$.
\item{C)}  Why doesn't this example contradict the 
Existence and Uniqueness Theorem?
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\noindent
II.  A student starting college places
\$ 500 into a bank account that pays interest
at the rate of 4\% per year, compounded continuously.
The student also saves \$ 520 over the course of each
year from her job and has it deposited into the account 
continuously.  Writing $B(t)$ for the balance in the account 
as a function of $t$ in years, these conditions
say 
$$B' = (.04)B + 520$$
What will the balance in the account be at the end of
4 years?   
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\noindent
III.  Construct the bifurcation diagram for the family of ODE
$$y' = y^3 + ay^2$$
where $a$ is an arbitrary real parameter.  For which values
of $a$ do bifurcations occur?
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\noindent
IV. 
\item{A)} Show that if $V$ is an eigenvector of $A$
with eigenvalue $\lambda$, then $X(t) = e^{\lambda t} V$
is a solution of $X' = AX$.
\item{B)}  Find the solution of $X' = \pmatrix{4 & 2 \cr 1 & 3\cr}X$
satisfying $X(0) = \pmatrix{1 \cr 0\cr}$.
\item{C)}  Sketch the phase portrait for the system from B.
\item{D)}  Find the solution of $X' = \pmatrix{0 & 2\cr -2 & -1}X$
with $X(0) = \pmatrix{2 \cr 1\cr}$.
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\noindent
V.  For the family of 1st order systems
$$X' = \pmatrix{a & 1 \cr a & a\cr},$$
sketch the corresponding curve in the trace-determinant plane,
and discuss any bifurcations that occur.
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\noindent
VI.  A $3\times 3$ matrix $A$ has characteristic polynomial
$(1 - \lambda)^3$  What are the possible canonical
forms $C$ of $A$, how would you tell which canonical form
$A$ has, and what are the corresponding geneneral 
solutions of the systems $Y' = CY$ for each possible $C$?
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\noindent
VII.  Consider the forced oscillator equation
$$y'' + 5y' + 6y =e^{-t} + 4$$
\item{A)}  Compute the solution with $y(0) = y'(0) = 0$.
\item{B)}  What would change in your solution if the $e^{-t}$
term on the right was replaced by $e^{-2t}$.
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\noindent
VIII.  Consider the nonlinear first order system
$$\cases{x' = -2x + y & \cr
         y' = -y + x^2 & \cr}$$
\item{A)} Find all the equilibrium points.
\item{B)} Linearize at each equilibrium and classify
the equilibrium of the linearized systems as sources, 
sinks, saddles, centers, or degenerate.
\item{C)}  What can you say about the types of the 
equilibrium points of the non-linear system based on B?
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\noindent
IX.  Consider 
$$\cases{x' = 1 - y^2&\cr
         y' = x(2 - y) &\cr}$$
\item{A)}  Is this system Hamiltonian?  Why or why not?
\item{B)}  Show that there is a function $\mu(x,y)$ such 
that the system 
$$\cases{x' = \mu(x,y)(1 - y^2)&\cr
         y' = \mu(x,y) x(2 - y) &\cr}$$
is Hamiltonian.
\item{C)} Find a conserved quantity for this new system,
then show it is also conserved for the original system.
\bye  
