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\headline={ESP Math 427K \hfil Spring 1997}
\footline={\ifnum\pageno>1 \hfil Worksheet 14\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 14} 
\bigskip

\noindent{\bf Terms to know:} {\it homogeneous equation,
nonhomogeneous equation, corresponding homogeneous equation,
complimentary solution, particular solution}

\bigskip 

\item{1.}  {\bf Undetermined Coefficients.}  Find the general solution
to 
$$
y''+y=3\sin2x+x\cos2x
$$
Hint:  Assume that a particular solution has the form
$$
y=(ax+b)\sin2x+(cx+d)\cos2x,
$$
and solve for the constants $a$, $b$, $c$, and $d$.

\medskip 
\item{2.}  {\bf Limiting Behavior of C.C. Homogeneous Equations}  
\itemitem{a)} If $a$, $b$, and $c$ are positive constants, show that
all solutions of $ay''+by'+cy=0$ approach zero as $x\to\infty$.

\medskip
\itemitem{b)}  If $a>0$ and $c>0$, but $b=0$, show that the result of
part a) is no longer true, but that all solutions are bounded as
$x\to\infty$.

\medskip
\itemitem{c)}  If $a>0$ and $b>0$ but $c=0$, show that the result of
part a) is no longer true, but that all solutions approach a constant
that depends on the intial conditions as
$x\to\infty$.  Determine this constant for the initial conditions
$y(0)=y_0$, $y'(0)=y_0'$.

\medskip
\item{3.}  {\bf Limiting Behavior of C.C. Nonhomogeneous Equations}  
\item{}  Consider the differential equation
$$
ay''+by'+cy=g(x),
$$
where $a$, $b$, and $c$ are positive.

\medskip
\itemitem{a)}  If $Y_1(x)$ and $Y_2(x)$ are solutions, show that
$Y_1(x)-Y_2(x)\to0$ as $x\to\infty$.  What does this mean
geometrically?

\medskip
\itemitem{b)}  Is this true if $b=0$?  Why or why not?

\medskip
\itemitem{c)}  If $g(x)=d$, a constant, show that every solution
approaches $d/c$ as $x\to\infty$.  What happens if $c=0$?  What if
$b=0$ also?

\medskip
\item{4.}  Find the general solution to 
$$
y''+2y'+y=2e^{-x}.
$$

\medskip
\item{5.}  Find the general solutions to the following equations using
both the method of undetermined coefficients and variation of
parameters.
$$
\pt a x^2y''-2y=3x^2-1,\quad x>0\hskip1in
\pt b x^2y''-3xy'+4y=x^2\ln x,\quad x>0.
$$


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