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\begin{document}
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{\newpage
\clearpage
\samepage \begin{displaymath}y''(t)+6y'(t)+9y(t)=0 \hspace{.2in}(*)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y_2(t)=v(t)e^{-3t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y'_2=v'e^{-3t}-3e^{-3t}v 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y''_2=v''e^{-3t}-6v'e^{-3t}+9ve^{-3t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}(v''e^{-3t}-6e^{-3t}v'+9e^{-3t}v)+6(v'e^{-3t}-3e^{-3t}v) 
+6e^{-3t}v=0
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}v''e^{-3t} = 0 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y_2(t)=v(t)e^{-3t}=(At+B)e^{-3t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y_2=Ate^{-3t}+Be^{-3t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y(t)=C_1e^{-3t}+C_2te^{-3t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y''(t)+p(t)y'(t)+q(t)y(t)=0 \hspace{.2in} (**)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}f''(t)+p(t)f'(t)+q(t)f(t)=0
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y_2=v(t)f(t)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y'_2=v'f+vf' \hspace{.2in} y''_2=v''f+2v'f'+vf''.
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}v''f+2v'f'+vf''+p(v'f+vf')+qvf=0
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}v(f''+pf'+qf)+v''f+2v'f'+pv'f=0
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}v''f+2v'f'+pv'f=0
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}v''(t)=-v'(t)(2\frac{f'(t)}{f(t)}+p(t))
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}v=\int\frac{u(t)}{[f(t)]^2}dt \hspace{.2in} (***)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}u(t)=e^{-\int p(t)dt}\hspace{.2in}(****)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y_2=f(t)\int\frac{u(t)}{[f(t)]^2}dt 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y''-ty'+y=0
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}u(t)=e^{-\int (-t)dt}=e^{\int t dt}=e^{t^2/2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}v=\int\frac{e^{t^2/2}}{t^2}dt 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y_2(t)=t\int\frac{e^{t^2/2}}{t^2}dt  
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y(t)=C_1y_1(t)+C_2y_2(t)
\end{displaymath}
}

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\end{document}
