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3. Solve the initial value problem (where $t > 0$): ~~
$
ty'+y=\sin t, \ ~y(\tfrac \pi 2)=1. 
$

From $
ty'+y=\sin t 
$, one can immediately go to the product rule step:

~~~~~~~~~~~$(ty)' = sin t$ ~~~~~~~~~~~~Check product rule $(ty)' = ty' + y$.

If you didn't notice the product rule:

$
1y'+\frac{1}{t} y=\frac{sin t}{t}$

Integrating factor $u(t) = e^{\int \frac{dt}{t}} = e^{ln(t)} = t$

Multiply BOTH sides by integrating factor:

$
ty'+y=\sin t 
$

$(ty)' = sin t$

$\int (ty)'dt = \int sin t dt$

$ty = -cos(t) + C$.  %Thus $y = -\frac{cos(t)}{t} + \frac{C}{t}$

 Plug in initial value, $y(\tfrac \pi 2)=1$:  

 $(\tfrac \pi 2)(1) = -cos(\tfrac \pi 2) + C \Rightarrow  (\tfrac \pi 2)(1) = 0 + C$.  Thus $C = \tfrac \pi 2$

Hence $ty = -cos(t) + \tfrac \pi 2$ and $y = -\frac{cos(t)}{t}+\tfrac \pi {2t}$

NOTE:  normally I would take off for not simplifying a fraction within a fraction.  I didn't this time, but will in the future.

Answer:  $y = -\frac{cos(t)}{t}+\tfrac \pi {2t}$
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4.  Fill in the blank:



 \begin{enumerate}[a.]
\hr


\item  
A 50 gallon tank contains 3 grams of Kryptonite. 8 gallons of water containing 2 grams of Krypotonite is pumped into the tanke every minute, and 8 gallons of solution is drained from the tank every minute.  Assuming a thorough mixing, setup the initial value problem that describes this process.  {\bf  (Do not solve!)}

\vfill
Diff. eqn:   ${\underline{~\hskip 3.5in~}}$.  Initial Value:   ${\underline{~\hskip 1.5in~}}$. 

\hr
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\item

 Calculate the Wronskian $W(t^{-\frac 12}, t^2)=\rule{8cm}{0.4pt}$. 
\vskip 60pt\vfill
%\hr
%\vskip 10pt
%\item  The longest interval, on which the following problem has a unique solution:
%$(t^2-1)y'+y=2t, \quad y(0)= 4$
%\vskip 1cm
%is $\noindent\rule{7cm}{0.4pt}$ (Do not solve!)
%\vskip 1cm


\hr

\item  Give the {\bf form} of the particular (nonhomogenous) solution with undetermined coefficients for $$y''+5y'+4y = 2\text{cos}(t)$$
\vskip 1cm
 $Y(t)=\noindent\rule{10cm}{0.4pt}$ (Do not solve!)


\hr  \vskip 20pt
\item The general solution for $y''-y=0$ is $\noindent\rule{9cm}{0.4pt}$
\vfill


\hr  \vskip 20pt
\item The general solution for $y''+9y=0$ is $\rule{9cm}{0.4pt}$. 
\vfill
 \end{enumerate}
\eject
~
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The longest interval on which the initial value problem 
\begin{align*}
\begin{cases}
(t^2-4t)y''+ty'+(t-4)y=e^t, \\
y(1)=1, y'(1)=2
\end{cases}
\end{align*}
 is $\noindent\rule{8cm}{0.4pt}$. 




\vspace{1in}





















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Last semesters exam (2021?)

\eject
\begin{enumerate}
\item  Draw a direction field for the differential equation ~~$ \;\;y'=(y+2)(4-y).$

Identify all equilibrium solutions and identify their stability.  On this direction field, also draw the solution that passes through the initial value $y(2) = 1$.  For this solution that passes through the point (2, 1), determine the behavior of $y$ as $t$ goes to $\infty$.

\np

\item  Find all the solutions defined in the interval $ (1, \infty)$ for the following differential equation: ~~ $\displaystyle{(t-1)y'=\frac{y^2}{t}}.$


%\vfill

%\centerline{Solution: $\underline{~~~\hskip 5in~~~}$}

\np

%\eject

\item  Find all solutions to the following differential equation:
\vskip -15pt
$$y'' + 4y = 2\sin(2t).$$


%\vfill\vfill\vfill\vfill\vfill\vfill\vfill

%\centerline{Solution: $\underline{~~~\hskip 5in~~~}$}
\np

%\newpage

\item  Find all the solutions of
%Find the solution for the following initial value problem % differential equation problem
$$y{''}+4y{'}+5y=0.$$%,\; y(0)=2,\; y{'}(0)=1.$$

\np
%\newpage

%$\cdot$




\item Solve the initial value problem:
$$y\,^\prime+\frac{3}{t}y=20,\; y(1)=2.$$
%\end{problem}
\np


\item%\begin{problem}{1}
A tank contains 4 gallons of water and  it contains 1 mg of salt
at  time $t=0$.  Fluid containing 2 mg per gallon of salt flows in at a rate of half gallon per minute and a well stired mixture flows out at the same rate.
Write the initial value problem of a differential equation to describe the amount $Q(t)$ of salt in the tank and find $Q(t)$ to the equation. What is the amount of salt in the tank after a long time?
%\end{problem}
\end{enumerate}

From Xiaoyi: 

\begin{enumerate}















\end{enumerate}

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