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\begin{document}

\noin Final Examination\hfill December 12, 2000\hfill 189-325A

\sk
\begin{enumerate}
\item
\begin{enumerate}
\item Find all solutions of the differential equation
$$2xy'-y^2+1=0$$
and sketch the solution curves.
\item For what pairs $(a,b)$ does the initial value problem
$$2xy'-y^2+1=0,\quad y(a)=b,$$
have (a) a unique solution, (b) more than one solution, (c) no
solution?
\end{enumerate}

\item Solve the initial value problem
$$(x^3+1)y'+x^2y-2xy^2-1=0,\quad y(0)=-1,$$
using the fact that there is an integrating factor which is a
function of $x-y$.

\item Find all solutions of the differential equation
$$y'''-y''+4y'-4y=\sin(x)+e^x.$$

\item Solve the initial value problem
$$
\ba{rl}
\f{dx}{dt}&=-4x-2y-1,\\
\f{dy}{dt}&=x-y-2, \ea
$$
with $x(0)=1,y(0)=-1$. What is the equilibrium solution?

\item Using Laplace transforms, solve the initial value problem
$$
y''+3y'+2y=\left\{\ba{l}
0, \quad 0\le t<1,\\
1, \quad 1\le t< \pi,\\
\sin(t), \quad \pi\le t, \ea\right.
$$
with $y(0)=1,y'(0)=-1$.

\item
\begin{enumerate}
\item Show that $x=0$ is a regular singular point of the differential
equation
$$x^2y''+(x^2-x)y'+y=0,\quad (x>0).$$
Find the indicial equation for this regular singular point.
\item Find the general solution of the above differential equation.
\end{enumerate}
\end{enumerate}
\end{document}
