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\begin{document}
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{\newpage
\clearpage
\samepage \begin{displaymath}y''-5y'+6y=0 \hspace{.2in}y(0)=2, y'(0)=2
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L[y''-5y'+y](s)=L[0](s)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L[y''-5y'+6y](s)=L[y''](s)-5L[y'](s)+6L[y](s)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L[y']=sL[y]-y(0)=sY(s)-2
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L[y'']=s^2L[y]-sy(0)-y'(0)=s^2Y(s)-2s-2
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L[y''](s)-5L[y'](s)+6L[y](s)=s^2Y(s)-2s-2-5sY(s)+10+6Y(s)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}(s^2-5s+6)Y(s)-2s-2+10=0
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{2s-8}{s^2-5s+6}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{4}{s-2}+\frac{-2}{s-3}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L^{-1}[1/(s-2)](t)=e^{2t} \hspace{.2in} L^{-1}[1/(s-3)](t)=e^{3t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y(t)=L^{-1}[\frac{4}{s-2}+\frac{-2}{s-3}](t)=4e^{2t}-2e^{3t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y''+4y'+5y=e^t \hspace{.2in}y(0)=1 y'(0)=2
\end{displaymath}
}

{\newpage
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\samepage \begin{displaymath}L[y''+4y'+5y]=L[10e^t]
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L[y''+4y'+5y]=L[y'']+4L[y']+5L[y]=s^2Y(s)-sy(0)-y'(0)+4sY(s)-4y(0)+5Y(s)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L[20e^t]=\frac{10}{s-1}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}(s^2+4s+5)Y(s)-s-6=\frac{10}{s-1}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{s+6}{s^2+4s+5}+\frac{10}{(s-1)(s^2+4s+5)}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{1}{(s+2)^2+1}+\frac{1}{s-1}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y(t)=e^{2t}\sin t+e^t.
\end{displaymath}
}

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\end{document}
