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\title{\vspace*{-1.5in}\textbf{\S7.3: Separable Differential Equations}}
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%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\chead{April 11, 2018}
\rhead{\S7.3: Separable Diff. Eq.}
\vspace{-.8in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item A \textbf{separable differential equation} is a differential equation that can be written in the form \[f(y)\cdot \frac{dy}{dx} = g(x).\]
\item To solve a separable differential equation:
\begin{enumerate}[1.]
\item Separate the variables
\item Integrate both sides (Remember ${}+ C$!!!)
\item Solve for $y$ (if possible)
\item Use the initial condition to find $C$.
\end{enumerate}
\item Other notes: \vskip2in
\end{itemize}


\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item Solve the differential equation $\frac{dy}{dx} = -2y$ if $y(0) = 1$. \vfill
\item Solve the differential equation $\frac{dx}{dt} +x = 1$ if $x(1) = 0.1$. \vfill
\newpage
\item Solve the differential equation $\frac{du}{dt} = u + ut^2$ if $u(0) = 5$. \vfill
\item Solve the differential equation $\frac{dy}{dx} = xe^y$ if $y(0) = 0$. \vfill
\item Solve the differential equation $\frac{ds}{d\theta} = -s^2\tan\theta$ if $s(0) = 2$. \vfill\newpage
\item Find an equation of the curve that passes through the point $(0,1)$ and whose slope at $(x,y)$ is $xy$.\vfill
\item Solve the differential equation $y' = x+y$ by making the change of variable $u=x+y$.\vfill
\newpage
\item A tank contains 20 kg of salt dissolved in 5000 L of water. Brine that contains 0.03 kg of salt per liter of water enteres the tank at a rate of 25 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt remains in the tank after half an hour? \vfill\vfill
\end{enumerate}
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