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\title{ Homogeneous, linear, second-order equations}
\date{February 22, 2021}

\begin{document}
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\vspace{-.2in}

\maketitle

\vspace{-.2in}

\dfn{
The \textbf{characteristic polynomial} of the differential equation $ay'' + by' + cy = 0$ is $ar^2 + br + c$ and the \textbf{characteristic equation} is $ar^2 + br + c = 0$.
}

\thm{
If the characteristic polynomial of $ay'' + by' + cy = 0$ has...
\begin{enumerate}[a)]
\item ...distinct real roots $r_1$ and $r_2$, then a general solution is $\displaystyle y = c_1e^{r_1 t} + c_2e^{r_2 t}$
\item ...a single (repeated) real root $r$, then a general solution is $\displaystyle y = e^{rt}(c_1 + c_2 t)$
\item ...complex roots $\lambda \pm i \omega$ (where $\omega > 0$), then a general solution is $\displaystyle y = e^{\lambda t}(c_1 \cos \omega t + c_2 \sin \omega t)$
\end{enumerate}
}


\prob{[Graded]
Solve the IVP: $y'' -2 y' + y = 0$, $y(0) = 2$, $y'(0) = -1$.
}

\prob{[Graded]
Solve the IVP: $6y'' - y' - y = 0$, $y(0) = -1$, $y'(0) = 3$.
}

%\newpage

\prob{[Completion]
An object stretches a spring $0.5$ feet at equilibrium.
You then compress the spring 4 inches and release it with an initial velocity of 1.5 ft/sec.
There's no damping.
Your goal is to find a formula for the motion of the spring.
\begin{enumerate}[a)]
\item Use the equilibrium information ($-mg  + \Delta l k = 0$) to solve for $k/m$. 
Note that you won't be able to solve for $k$ or $m$ individually (and you don't need to).
\item Use your value for $k/m$ to find a general solution to the differential equation $my'' + ky = 0$.
\item Use the two initial conditions (for $y(0)$ and $y'(0)$) to find the values of the two constants.
\end{enumerate}
}

\noindent\hrulefill

\vspace{.1in}

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The general solution to the undamped spring system $my''+ky = 0$ is $\displaystyle y= c_1 \cos\left( \sqrt{\frac{k}{m}} t \right) + c_2 \sin\left( \sqrt{\frac{k}{m}} t \right) $.
Most of the world prefers to express undamped oscillation as $\displaystyle \boxed{ y = R \cos(\omega_0 t - \phi)}$ where 
\begin{itemize}
\item $\displaystyle \omega_0 = \sqrt{\frac{k}{m}}$
\item $\displaystyle R = \sqrt{c_1^2 + c_2^2}$
\item $\phi$ is an angle such that $c_1 = R \cos(\phi)$ and $c_2 = R \sin(\phi)$
\end{itemize}

\prob{[Completion]
Express your solution to the last problem in the form $y = R \cos(\omega_0 t - \phi)$ (that is, find the values of $R$, $\omega_0$, and $\phi$ for your solution).
}

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