
% extra credit ch1 2280, S2016
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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2280 Extra Credit Problems \\
\bf Chapter 1 \\
{\small\bf S2016}
\end{center}


{\bf Submitted work}. Please submit one stapled package per chapter.
Kindly label problems \fbox{\bf Extra Credit}. Label each problem with
its corresponding problem number, e.g., \fbox{Xc1.2-4}. Please attach
this printed sheet to simplify your work.

\Prob{Xc1.2-4}{(Quadrature)}
Solve $y'=x^{-2}+x^{-1}$, $y(1)=2$.

\Prob{Xc1.2-10}{(Quadrature)}
Solve $y'=xe^{-2x}+x^2$, $y(0)=2$.

\Prob{Xc1.3-8}{(Picard's theorem)}
Find a box with center $x=0$, $y=0$ to which Picard's theorem applies,
verifying also continuity of $f(x,y)$ and $f_y(x,y)$ in the box, for the equation
$$y'=x\sqrt{x+y+1},\quad y(0)=0.$$

\Prob{Xc1.3-14}{(Peano's theorem)}
Does Peano's theorem apply to establish existence of at least one solution, for the problem below?
Please carefully check the hypothesis of the theorem, which is continuity of $f(x,y)$ on a box with center $x=0$, $y=1$.
$$
 y'=3(y-1)^{1/3},\quad y(0)=1.
$$


\Prob{Xc1.4-6}{(Separable DE)}
Solve for equilibrium and non-equilibrium solutions (find the general solution).
$$y'=2x\sec y.$$

\Prob{Xc1.4-17}{(Separability test)}
Use the test to verify that the equation $y'=e^x+e^y$ is not separable.

\Prob{Xc1.4-18}{(Separability test)}
Find a factorization $f(x,y)=F(x)G(y)$ for the problem below and then
determine all non-equilibrium solutions.
$$
y' = x^2(y^2+y)+y^2+x^2y+2y+x^2+1.
$$

\Prob{Xc1.4-49}{(Newton cooling)}
A roast is put into an oven whose temperature is $400^\circ$ F.
The meat thermometer was initially at $40^\circ$ F and after 30 minutes
it rose to $90^\circ$ F. The roast is done when the thermometer reaches
$340^\circ$ F. How long does it take to cook the roast?


\Prob{Xc1.5-4}{(Linear DE)}
Solve $y'-2xy=e^{x^2}$.

\Prob{Xc1.5-16}{(Linear DE)}
Solve $y'=(1-y)\cos x$, $y(\pi)=2$.

\Prob{Xc1.5-24}{(Linear DE)}
Solve $(x^2+4)y' + 3xy = x$, $y(0)=1$.

\Prob{Xc1.5-38}{(Brine tank)}
Solve the brine tank problem
$$
\begin{array}{lcrcl}
 x'(t) &=& -4x(t), \\
 y'(t) &=& 4x(t) &-& 5y(t).
\end{array}
$$

{\bf End of extra credit problems chapter 1.}

\end{document}

