Demonstration 1: Direction Fields.

Programming

\(
<< Graphics`PlotField`\)Clear[PlotDirectionField]\(PlotDirectionField[f_, \ {x_, \ xmin_, \ xmax_}, 

        	{y_, \ ymin_, \ ymax_}, 
	\ opt___]\  := 
	
      PlotVectorField[
        \ {Sqrt[1/\((f^2\  + \ 1)\)], 
			f*Sqrt[1/\((f^2\  + \ 1)\)]}, 
        \ {x, \ xmin, \ xmax}, 
		{y, \ ymin, \ ymax}, Frame -> \ True, 
        opt]\)

The basic syntax for plotting a direction field  f(y,t)  is 
PlotDirectionField[ f[y,t],  {t, tmin, tmax}, {y, ymin, ymax}].  
The options PlotPoints -> n   and ScaleFactor ->  s can be used 
to adjust the appearance of the plot.  Experiment  with their effect!

\(\(
PlotDirectionField[Cos[t], \ {t, \ \(-10\), \ 10}, \ 

      		{y, \ \(-5\), \ 5}, 
	PlotPoints -> 20, \ \ ScaleFactor -> \ .75]
      \)\)

\(\(
PlotDirectionField[y + 3, \ {t, \ \(-1\), \ 5}, \ 

      		{y, \ \(-5\), \ 1}, ScaleFactor -> .3
	]\)\)

\(\(
PlotDirectionField[7  y + 3  t, \ {t, \ \(-1\), \ 5}, \ 

      		{y, \ \(-3\), \ 3}, 
		PlotPoints -> 20, \ \ 
      ScaleFactor -> \ .25]\)\)

\(PlotDirectionField[\ \((y/2\  + \ Exp[\(-x\)])\), 
	{x, \ 0, \ 5}, 
      \ {y, \ \(-3\), \ 3}\ , 
	ScaleFactor\  -> \ .3]\)

\(PlotDirectionField[\(-3\) y\  + \ t\  + \ Exp[\(-2\) t], 

      	{t, \(-2\), 7}, \ {y, \(-2\), 4}, \ ScaleFactor\  -> \ .3]\)