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%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large Math 148 Assignment 7}\\[1ex]
\textbf{Due  2:00 p.m.\ Wednesday, April 1 in the Math 148 dropbox.}  
\end{center}

\vskip 1em 
\begin{Ex}

\item 
\begin{parts}
\item  Solve the DE $(x^2-y^2)y' - 2xy = 0$. 
\item Sketch the set of solution curves.
\item If $(1,2)$ is a point on the solution curve, find the solution.
\end{parts}


\item 
\begin{parts}
\item Solve the DE $(1+x^2)y' + 2xy = 0$.                                                                                          
\item Solve the DE $(1+x^2)y' + 2xy = \cot x$. 
\end{parts}


\item 
\begin{parts}
\item Consider the DE $y' + p(x)y = q(x) y^a$ where $a \not\in\{0,1\}$ is a real number.\\ 
Set $z = y^{1-a}$. Turn this DE into a linear DE in $z$.
\item Use this to solve $xy^2y'+y^3 = x\cos x$.
\end{parts}



\item Torricelli's Law for fluid flowing out the bottom of a tank states that the velocity is
calculated as if the fluid dropped from the surface of the water. 
A hemispherical container of radius $R$ is completely full of water. 
A small round hole of radius $r$ at the bottom is unplugged.
(You can ignore the small difference in the height caused by the hole.)
\begin{parts}
\item Use Newtonian mechanics to deduce how long it takes a drop of water to fall distance $h$ from a position of rest,
and use this to compute the velocity at that point.
\item Compute the volume $V(h)$ in the bowl when the water depth is $h$.
\item How long does it take the bowl to empty?\\ 
\hint compute $\frac{dV}{dt}$ in two ways. Get a DE in $h$.
\end{parts}



\item  Consider the homogeneous linear DE $y'' + p(x)y' + q(x) y = 0$.
Suppose that $y_1$ and $y_2$ are two solutions on $[a,b]$.
Define $W(x) = y_1(x) y_2'(x) - y_1'(x) y_2(x)$.
\begin{parts}
\item Find a first order DE satisfied by $W$ and solve it.
\item Prove that if the vectors $(y_1(c), y_1'(c))$ and $(y_2(c), y_2'(c))$ are linearly independent for some $c\in [a,b]$,
then $W(x)$ never vanishes. Hence show that $(y_1(x), y_1'(x))$ and $(y_2(x), y_2'(x))$ are linearly independent vectors
for every $x \in [a,b]$.
\end{parts}


\item Consider a curve given in polar coordinates by $r(\theta) = \dfrac1{1 + e \cos\theta}$, where $e\ge0$.
\begin{parts}
\item  Show that the distance of each point on this curve to the line $x=\frac1e$ is a constant multiple of $r(\theta)$.
\item  When $e>1$, show that the curve approaches two asymptotes, find them and sketch the curve. \hint If the critical angles are $\pm\theta_0$, compute the vertical distance of the point of the curve at angle $\theta = \theta_0+h$ to the line $\theta=\theta_0$, and take a limit using Taylor approximations.
\item Observe that the curve is bounded if and only if $e < 1$. Show that the curve is an ellipse as follows: Let $a$ be the midpoint between the two points intersecting the $x$-axis. Show that $(1-e^2)(x-a)^2 + y^2$ is constant.
\item What happens when $e=1$?
\end{parts}

\end{Ex}
\end{document}
