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22M:100 (MATH:3600:0001) Exam 2 \hfil \break
April 27, 2013


1.)  Circle T for True and F for false.


[4]~ 1a.)  If $y = f(t)$ and $y = g(t)$ are solutions to $y'' + p(t)y' + q(t)y = 0$, then $y = c_1f(t) + c_2g(t)$ is also a solution to this differential equation for any constants $c_1$ and $c_2$.

\rightline{T \hskip 0.8in F}

\v

[4]~ 1b.)  If ${\bf x} = {\bf f}(t)$ and ${\bf x} = {\bf g}(t)$ are solutions to ${\bf x}' = A{\bf x}$, then ${\bf x} = c_1{\bf f}(t) + c_2{\bf g}(t)$ is also a solution to this system of differential equation for any constants $c_1$ and $c_2$.

\rightline{T \hskip 0.8in F}

\v\v

[12]~ 2.)  The phase portrait for  $\frac{dx}{dt} =x(y-2) $ and $\frac{dy}{dt} = x + 3$
is drawn below. Find all equilibrium solutions and determine whether the critical point is asymptotically stable, stable, or unstable.  Also classify it as to type (nodal source, nodal sink, saddle point, spiral source, spiral sink, center).


\vfill
\centerline{Equilibrium solution: $\underline{\hskip 3in}$}
\v\v

\centerline{~~~Stability : $\underline{\hskip 3in}$ \hfill Type: $\underline{\hskip 3in}$~~~}


\eject

[20]~ 3.) Find a recursive formula for the constants of the series solution to
$y'' + 2y = 0$ near $x_0 = 0$.

\vfill
\centerline{Answer: $\underline{\hskip 3in}$}
\eject
[20]~ 4.) Suppose that $\mathlarger{a_{2k} = \frac{-a_{2k-2}}{4k^2}}$.  Use induction to prove that
$\mathlarger{a_{2k} = \frac{(-1)^ka_0}{4^k(k!)^2}}$ for all $k \geq 0$.

\eject

[20]~ 5.) Solve ${\bf x}' = \left(\matrix{0 & 3 \cr 3 & 0}\right){\bf x}$

\vfill
\centerline{Answer: $\underline{\hskip 3in}$}
\eject

[20]~ 6.)  Suppose the matrix $A$ has eigenvalue -1 with eigenvector $\left(\matrix{0  \cr 3}\right)$ and \hfil \break
eigenvalue 4 with eigenvector $\left(\matrix{2  \cr 1}\right)$.  Draw the phase portrait for this system of differential equations.  
Identify the equilibrium solution.  Also state the general solution.
\v\v

Equilibrium solution is $\underline{\hskip 3in}$
\v\v
General solution is $\underline{\hskip 3in}$
\v
\includegraphics[width = 5.2in]{grapht.eps}  %% export as pdf file 


 

 
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