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Cell[CellGroupData[{
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 StyleBox["MAT 303:",
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 " ",
 StyleBox["Homework 3\n",
  FontSize->24],
 StyleBox["due Monday, September 26",
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Cell["\<\
In these exercises, we want to use Mathematica to visualize solutions of \
first order differential equations and gain further insight into their long \
term behavior. Please read the notes posted on the Course Webpage before \
solving these exercises.\
\>", "Text",
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Cell[TextData[{
 " Study the differential equation ",
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 "Plot the vector field using ",
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 " in the region x \[Element] [-3.5, 3.5], y \[Element] [-3.3, 3.3]. Use an \
option of ",
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 "Plot the vector field using ",
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  FontWeight->"Bold"],
 "in the same region x \[Element] [-3.5, 3.5], y \[Element] [-3.3, 3.3]. On \
the same plot, highlight the solution curve passing through the point (0, 1) \
in Red, and the solution (integral curve) passing through the point (1, -1.5) \
in Green. "
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Cell[TextData[{
 "Explain mathematically the discontinuities of the vector field (where the \
direction of the arrows changes, and the slopes of the integral curves are \
\[Infinity] or -\[Infinity]). Then find numerically the places where these \
discontinuities occur using ",
 StyleBox["NSolve[expr,vars]. ",
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The example below shows an application of NSolve (notice the double equal \
sign == that is used to test equality between expressions).\
\>", "Item",
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Cell[TextData[{
 "The ODE given in Problem 3.1 is a separable differential equation, that we \
can solve and get the family of solutions defined by the implicit equation  \
",
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 " . Plot together on the same picture the implicit solutions corresponding \
to the following choices of the constant ",
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 ". Is the picture consistent to the Vector Field pictures obtained in part \
",
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 "The function ",
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below show an application of ContourPlot.  "
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 "Plot the vector field using ",
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 " in the region x \[Element] [-5, 5], y \[Element] [-5, 5]. On the same \
vector field, draw the integral curve corresponding to the initial condition \
(2,1). What do you notice?"
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Cell["\<\
by the Integrating Factor Method, noting the fact that this is a linear \
equation.\
\>", "SubitemNumbered",
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Cell["\<\
by the Substitution Method, by doing the substitution v = x-y.\
\>", "SubitemNumbered",
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Cell[TextData[{
 "Define a function in ",
 StyleBox["Mathematica, ",
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 "called ",
 StyleBox["sol[x_,c_] ",
  FontWeight->"Bold"],
 "that outputs the solution that you have computed at part 2. Use the \
function Plot[...] to plot togther several solutions of the differential \
equation."
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Explain the presence of a linear solution y(x)=x-1 of the differential \
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Analyze the local and global existence and uniqueness of solutions of the \
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obstruction to global uniqueness of solutions (in other words, can it be used \
to produce infinitely many solutions)?\
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