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Math 305 \hfill Homework Set 7  \hfill Spring 2017

\begin{center}Due Monday March 6 \end{center}

{\bf I.} For each differential equation below, find the general 
solution. 


\begin{enumerate}
\item $y'' + y = \csc t$, $0<t<\pi$.

\item $y'' + y = \sec t \tan t$, $-\pi/2 < t < \pi/2$.

\item $\frac{d^2y}{dx^2} - \frac{dy}{dx} = x^3$

\item $y'' + 4y' + 5y = 10$.

\item $x^2 y'' +x y' - y = x$.

\item $y' = -\frac{3x^2 + y}{x+2y}$

\item $4y'' + 4y' + y = e^{-t/2}$

\item $y' + ty = ty^3$


\end{enumerate}


{\bf II.}1. A 9 lb object is attached to the lower end of a spring whose upper 
end is attached to the ceiling. The spring constant $k$ is 1 lb/ft. The resistance 
to the motion is $\gamma =0.25$ lb-s/ft times the velocity of the object. The object is set in motion 
by pulling it down 1 ft and then letting it go.

Set up a differential equation to model this mass-spring system. Solve it and 
plot the solution. (Don't forget to convert weight to mass.)


2. A 9 lb object is attached to the lower end of a spring whose upper end is 
attached to the ceiling. The spring constant $k$ is 1 lb/ft. The resistance to 
the motion is $\gamma =0.25$ lb-s/ft times the velocity of the object. The system is in equilibrium 
(no motion). Then, at time zero, a 1 lb hawk lands gently onto the object. Now what happens? 

Set up a differential equation to model this mass-spring system. Solve it and 
plot the solution. 


3. A 9 lb iron object is attached to the lower end of a spring whose upper end is 
attached to the ceiling. The spring constant $k$ is 1 lb/ft. The resistance to the 
motion is  $\gamma =0.25$ lb-s/ft times the velocity of the weight. The system is in equilibrium (no motion). 
Then, at time zero, a oscillating magnet on the floor is turned on. It exerts a force on the 
object of $2 \sin t$ pounds - positive is down and negative is up. 

Set up a differential equation to model this mass-spring system with external 
forcing. Solve it and plot the solution. 

{\bf III.} Essay question: Look up non-linear springs. Describe several types and 
what they might be used for. List references you use. Where would I buy one? 

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