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{\bf Question}

Find the general solution for each of the following second order
equations:
\begin{description}
\item[(a)]$y''-3y'+2y=0$
\item[(b)]$y''+4y'+13y=0$
\item[(c)] $y''-4y'+y=0$
\item[(d)]$y''-3y'=0$
\end{description}



\vspace{.5in}

{\bf Answer}

\begin{description}
\item[(a)]
\begin{eqnarray*} y'' - 3y' + 2y & = & 0 \\ {\rm A.E. \ \ }
m^2 - 3m + 2 & =& 0 \\ (m-1)(m-2) & =& 0 \\ m = 1 {\rm \ \ or\ \ }
m = 2 \\ {\rm so\ \ } y & =& Ae^x +  B e^{2x} \end{eqnarray*}
\item[(b)]
\begin{eqnarray*} y''+ 4y' + 13y & = & 0 \\ {\rm A.E. \ \ }
m^2 + 4m + 13 & =& 0 \\ (m+2)^2 & =& -9 \\ m = -2 \pm 3i \\ {\rm
so\ \ } y & =& e^{-2x}(A \cos 3x + B \sin 3x)
\end{eqnarray*}
\item[(c)]
\begin{eqnarray*} y'' - 4y' + y & = & 0 \\ {\rm A.E. \ \ }
m^2 - 4m + 1 & =& 0 \\ m =  2 \pm \sqrt 3
\\ {\rm so\ \ } y & =&  e^{2x}(Ae^{\sqrt 3 x} + Be^{-\sqrt 3 x}) \end{eqnarray*}
\item[(d)]
\begin{eqnarray*} y'' - 3y' & = & 0 \\ {\rm A.E. \ \ }
m^2 - 3m & =& 0 \\ m(m-3) & =& 0 \\ m = 0 {\rm \ \ or\ \ } m = 3
\\ {\rm so\ \ } y & =& A +  B e^{3x} \end{eqnarray*}
\end{description}


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