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\centerline{Mathematics 304 -- Ordinary Differential Equations}
\centerline{Discussion 1 -- Qualitative Techniques for 1st Order ODE}
\centerline{August 31, 2000}
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{\it Background}
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For a first order ODE ${dy\over dx} = f(x,y)$, the {\it direction field}
gives valuable qualitative information about solutions.  The direction
field is a plot in the $(x,y)$-plane, where a small line segment of 
slope $f(x,y)$ is placed at the point $(x,y)$.
Conceptually, we do this at all points where $f(x,y)$ is defined,
but that is impossible to draw, so some finite collection of 
``representative'' points $(x,y)$ is used in practice.  

To draw slope fields by hand, a useful technique is:
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\item{1.}  First find the {\it isoclines} of the equation -- the curves
of constant slope value given by the equations $f(x,y) = c$.  
(These are the same as the {\it level curves} of the slope
function $f(x,y)$ as defined in multivariable calculus.)  
\item{2.}  Sketch some collection of the isoclines (often the isoclines
for integer slope values, e.g.
$$c = \ldots, -2,-1,0,1,2, \ldots$$ 
are sufficient to see what is happening)
\item{3.}  Draw in line segments of slope $c$ at points on the 
isocline for the slope value $c$.
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Then, for instance, we can often get qualitative information about the 
solutions, since a solution $y = \phi(x)$ is a function whose graph
``follows the direction field'' -- at each point on the graph,
the slope of the tangent line satisfies
$$\phi'(x) = {dy\over dx} = f(x,\phi(x)).$$

From this picture, the following should be plausible:
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\proclaim Existence and Uniqueness Theorem for ODE Initial Value Problems.
Let $f(x,y)$ and ${\partial f(x,y)\over \partial y}$ be defined and 
continuous on a rectangle $R$ in the plane containing $(x_0,y_0)$.  Then
there is a unique solution $\phi(x)$ of the IVP
$$\cases{{dy\over dx} = f(x,y)&\cr
              y(x_0) = y_0&\cr}$$
defined on some open interval containing $x_0$.\par
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Today, in our first discussion, we will study direction fields for
some differential equations and see what they tell us about solutions.

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{\it Discussion Questions}
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\item{A)}  Sketch the isoclines with $c = -2, -1, 0, 1, 2$ for the
differential equation 
$${dy \over dx} = y - x$$
and use them to sketch the direction field.  What can you say
about the function $\phi(x) = x + 1$ (is it a solution of this equation?)  
Give rough qualitative sketches of the solutions $\phi(x)$ with 
$\phi(0) = 0$ and $\phi(-2) = 1$.
Describe the graphs of solutions of the initial value problems:
$$\cases{{dy \over dx} = y - x&\cr
                 y(-1) = \alpha&\cr}$$
What happens as $x \to +\infty$?  How does your answer depend on the value
of $\alpha$?

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\item{B)}  A first order ODE whose slope function is {\it independent of $x$}
(that is, which is of the form ${dy \over dx} = f(y)$)
is called an {\it autonomous} equation.
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\itemitem{1)}  What do the isoclines of an autonomous
equation look like?  (What kind(s) of curve(s) are they?)
\itemitem{2)}  Suppose that $f,f'$ are continuous on all of ${\bf R}$.
If $x_0$ is an arbitrary $x$-value, show that the Existence
and Uniqueness Theorem for 1st order ODE 
implies that there is one and only one solution of the IVP 
$$\cases{{dy \over dx} = f(y)&\cr
            y(x_0) = y_0&\cr}$$
Explain why this says that the graphs of two different solutions of an 
autonomous equation {\it never intersect}.
\itemitem{3)}  Suppose that $f(y_0) = 0$ for some particular $y_0$.
Explain why these are called {\it equilibrium points} for the ODE.
What can you say about the constant function $\phi(x) = y_0$?
(Is it a solution of the equation?)
\itemitem{4)}  For each of the following autonomous ODE, find 
all the equilibrium points, sketch the part of the direction field 
for $-4 \le y \le 4$ and a 
``representative sampling'' of solutions:
$$\eqalign{{dy\over dx} &= -y^2 + 5y - 6 = (y - 2)(3 - y)\cr
           {dy\over dx} &= y^3 - 2y^2 + y = y (y-1)^2 \cr
           {dy\over dx} &= y\cos(2y)}$$
\itemitem{5)}  One important qualitative question about solutions
of ODE is:  {\it what is their long-term behavior}?  That is, 
what is $\lim_{x \to \infty} \phi(x)$?  In this part, you 
will study how the graph of 
the slope function (i.e. the graph $w = f(y)$) can help determine
this.  In particular, when is it true that all solutions starting 
from points near $y = y_0$ tend toward $y = y_0$ as $x \to \infty$?
When is it true that all solutions starting near an equilibrium move 
away?  Is the behavior always the same on both sides of an equilibrium?
Use your work from part 4 to help formulate a general statement about
how the graph $w = f(y)$ relates to the behavior of solutions near
equilibria, and justify your statements.
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{\it Assignment}
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Group write-ups due in class on Tuesday, September 5.
\bye
