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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Discussion 5 -- Differential Equations and Slope Fields}
\centerline{April 4, 2001}
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{\it Background}
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In class on Tuesday, we introduced the idea of a {\it differential equation} -- 
a relation between derivatives of an unknown function, that function,
and other known functions of one variable.  The {\it order} of a 
differential equation is the highest order derivative of the unknown
function that appears.  For instance, 
$${d^2y\over dt^2} + 5{dy\over dt} + 6y = \cos(4t)$$
is a second order differential equation for the unknown function
$y$.  Today we want to practice recognizing when a function solves
a differential equation.  

We will also introduce a {\it graphical way}
to understand how the graphs of solutions of a general first order 
differential equation ${dy\over dx} = f(x,y)$ are determined by the
equation.  This will use what is called the {\it slope field} of the 
equation.  The idea is the following.  If we have any function 
$y=y(x)$, then the value of the derivative ${dy\over dx}$ tells us the 
slope of the graph (i.e. the slope of the tangent line to the graph)
at each point.  Hence the differential equation ${dy\over dx} = f(x,y)$
says that in order for a function $y = y(x)$ to be a solution, 
{\it at each point $(x_0,y_0) = (x_0,y(x_0))$ on the graph
of the solution function, the slope
of the tangent line at that point has to be equal to the value we
get from the ``slope function'' $f$ at that point:  $f(x_0,y_0)$.}

Hence, to visualize the graphs of solutions, we can make a plot 
showing the slopes $f(x_0,y_0)$ at a collection of points.  (We 
usually use points in some grid or subdivision of a part of the 
$xy$-plane.)  The slope value at $(x_0,y_0)$
is indicated by putting in a small line segment of that slope
with its midpoint at $(x_0,y_0)$.  Then the graphs of solutions
can be visualized by {\it drawing in curves that ``follow'' the 
slope field -- that are tangent to the slope field line segments
at each point they pass through}.
\vglue 2.5truein
\centerline{Slope field for ${dy\over dx} = y\cos(\sqrt{x}\sin(x))$}
\centerline{and solution with $y(1) = 3$}
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{\it Discussion Questions}
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A)  Which of the following list of functions are solutions of which of the
list of differential equations.  (There may be zero, one, or more than
one solution for each of the differential equations.)
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\noindent 
Functions:
$$y_1(x) = \sin(4x)\qquad y_2(x) = \cos(4x) \qquad y_3(x) = e^{-4x}
\qquad y_4(x) = x^4\qquad y_5(x) = x^{-3}$$
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Differential Equations:
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\item{1)} ${d^2y\over dx^2} = -16y$
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\item{2)} ${dy\over dx} = -4y$
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\item{3)} ${dy\over dx} = 4y$
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\item{4)} $x^2{d^2y\over dx^2} - 12y = 0$
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B)  Problem 1 from Section 9.2 in our text
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C)  Problem 7 from Section 9.2 in our text
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{\it Assignment}
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Group writeups due in class on Tuesday, April 10.
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