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Math 305 \hfill Homework Set 2  \hfill Spring 2017

\begin{center}Due Monday January 30\end{center}

{\bf I.} Find the general solution to each of the separable differential equations below. If an initial condition is present also find the particular solution. These problems are based on Section 2.2.


\vspace{.15in}

\begin{enumerate}
\item $y' = y$ and $y(0)=-6$.

\item  $y' = y$ and $y(1) = e^2$.

\item  $y' = \cos(2x) + 2$ and $y(\pi) = 8$.

\item  $y' = y^2e^x$ and $y(0)=2$.

\item  $y' = e^{x+y}$.

\item  $y' = x^2\cos^2(y)$

\item  $y' = y^2 e^x$

\item  $y' = x \sin(x) \cot(y)$ and $y(\pi/4)=6$.

\item  $y' = x^3y^2 + xy^2 + 2y^2 + x^3 + x + 2$ and $y(0) = 1$.

\item  $y' = xy^3\sqrt{1+x^2}$ and $y(0)=1$.

\end{enumerate}


{\bf II.} Describe the set of initial conditions for which each differential equation below is guaranteed to satisfy Theorem 2.4.2. Do not try to solve them.

\vspace{.15in}


\begin{enumerate}
\item  $y' = \frac{x+2}{2-y}$

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

\item  $y' = \frac{y}{x-3} + \frac{x^2}{y+4}$

\item  $y' = \frac{\tan(x)}{y^2-1}$

\item  $y' = \cot(xy)$

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

\item  $y' = (xy)^{2/3}$

\end{enumerate}

{\bf III.} Find the general solution to each of the linear differential equations below. If an initial condition is present also find the particular solution. These problems are based on Section 2.1.

\begin{enumerate}
\item $y' + xy = x$.
\item  $xy' = x^2 + 3y$.
\item $e^x y' + 2e^x y = 1$.
\item  $3xy' - y = 1 + \ln x$, $y(1) = -2$. On what interval is your solution valid?
\item $y' + (\tan x) y = \cos^2 x$, $y(0) = 1$. On what interval is your solution valid?

 \end{enumerate}

{\bf IV.} Find the largest interval on which a solution to each linear initial value problem below 
must exist by Theorem 2.4.1. Do not try to find the solution.

\begin{enumerate}
\item $y' + \dis \frac{y}{t} = \frac{1}{t^2-1}$, $y(-0.5) = 4$.
\item $y' + \dis \frac{t^2 + 1}{t-14}y = \frac{\sin t}{t-4}$,  $y(5) = 14,000$.

\item $y' + \frac{1}{t^3-8} y = \frac{1}{t^2-9}$, $y(0)=4$.

\item $y' +\cot (t) y = tan (t)$, $y(\pi/6) = 0$.

\end{enumerate}

{\bf V.} Find the general solution to each of the differential equations below. Use the Bernoulli method. 

\begin{enumerate}

\item $y' - y = xy^2$

\item  $y' + \dis \frac{y}{x} = xy^2$.

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

\item $y'+ 3y = y^3 \sin x$.


\end{enumerate}

{\bf VI.} Draw the direction field by hand for the following differential equations. Use graph paper with grid locations from -5 to 5 for x and y. Do not solve these equations.

\begin{enumerate}
\item $y' = 2x-1$

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

\item  $y' = y+1$

\item  $y' = 2x+y$

\item  $y' = \frac{1}{3}(x^2-1)$
\end{enumerate}


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