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\centerline{Mathematics 132 -- Calculus for Physical and Life Sciences 2}
\centerline{Discussion 6 -- Slope Fields}
\centerline{April 22, 2005}
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\leftline{Your Group: $\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$}
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\noindent
A) Which of the following slope fields goes with which differential
equation? 
\vglue 5.0truein
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\noindent
Plot I: $\underline{\ \ \ \ \ \ \ }$ Plot II: $\underline{\ \ \ \ \ \ \ }$
Plot III: $\underline{\ \ \ \ \ \ \ }$ Plot IV: $\underline{\ \ \ \ \ \ \ }$
Plot V: $\underline{\ \ \ \ \ \ \ }$ Plot VI: $\underline{\ \ \ \ \ \ \ }$
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\noindent
B) You should have found that plot III corresponds to differential
equation (e) in part A.  
\item{1)}  On the plot of III, sketch the solutions passing through the points
$(0,1), (0,-1), (0,0)$ (you should have three different graphs).
\item{2)}  The exact formula for the solution of equation (e) or
slope field III with $y(0) = -1$ is $y(x) = x - 1$.  Verify that
this function does solve equation (e):
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\noindent
{\it Answer:}
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C)  To get numerical approximations to values of the solution 
$y = y(x)$ of a differential equation ${dy\over dx} = f(x,y)$
with given initial value $y(x_0) = y_0$, we could proceed as 
follows: 
\item{$\bullet$}  Pick a ``step size'' $h$ (like the $\Delta x$ for a Riemann 
sum).  
\item{$\bullet$}  Start at the point $(x_0,y_0)$, ``sample'' the slope
function there and go along a straight line of slope $f(x_0,y_0)$
until you get to $x_1 = x_0 + h$.  The $y$-coordinate on that line
with $x=x_1$ is $y_1 \doteq y(x_1)$.  
\item{$\bullet$} Now ``sample'' the slope
function at $(x_1,y_1)$ and go along a straight line of slope $f(x_1,y_1)$
until you get to $x_2 = x_1 + h = x_0+2h$.  The $y$-coordinate on that line
is $y_2 \doteq y(x_2)$.  
\item{$\bullet$}  Repeat the process as often as needed (for instance, 
until $x_k = $ some $x$ value where you want to know an approximate
solution value.
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The process described above leads to the equation
$$y_k = y_{k-1} + f(x_{k-1},y_{k-1}) h$$
for producing $y_k$ from $x_{k-1},y_{k-1},h$.  
For instance, if we are looking for numerical approximations
to the solution of ${dy\over dx} = 2y$ with $y(0) = 1$, we 
might choose $h = 0.2$.  Then $(x_0,y_0) = (0,1)$, 
$x_1=0.2$, and
$$y_1 = y_0 + f(x_0,y_0)h = 1 + 2(1)(.02) = 1.04$$

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Continue this process to approximate the solution of 
${dy\over dx} = 2y$ with $y(0) = 1$ at 
$x_2 = .4$, $x_3 = .6$, $x_4 = .8$, and $x_5 = 1.0$.  Present
your work as a table of values at these $x$'s.
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$$\matrix{ k & x_k & y_k\cr
         0 & 0   & 1\cr
         1 & .2  & \underline{1.04}\cr
         2 & .4  & \underline{\ \ \ \ \ \ }\cr
         3 & .6  & \underline{\ \ \ \ \ \ }\cr
         4 & .8  & \underline{\ \ \ \ \ \ }\cr
         5 & 1.0  & \underline{\ \ \ \ \ \ }\cr}$$
         
         
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\noindent
{\it Assignment}
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Group writeups due no later than Monday, April 25.
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