Autonomous Differential Equations      Click here for info on semi-stable solutions.            

 

Example:    Plot the slope field for:        dy/dt  =  (1/2) y ( y ˗ 2 )2 ( y ˗ 4 )  =  f(y)

 

Strategy:  Set up a table including values of y in each region and calculate the slopes, f(y).

 

Note: Values of slope are independent on  t.  So "solution curves" are the same for

translations in t.

 

For  y > 4              f(y)  for  4.25,  4.5,  4.75, 5.0,  etc                  calculate slope and plot

 

For  2  <  y  < 4     f(y)  for  2.25, 2.5,  3.0,  3.5,  3.75,  etc          calculate slope and plot

 

For  0  <  y <  2     f(y)  for  1.9,  1.75,  1.5, 1.0,  0.5,  etc            calculate slope and plot

 

For  y < 0              f(y)  for  ˗ 3.0,  ˗ 2.5,  - 2.0,  ˗ 1.5,  ˗ 0.5  etc   calculate slope and plot

 

 

 

                           

                                                 

                                  Plot of Slope Field Showing Solution Curves

           

Qualitative response for y(t) showing solution curves for stable, semi-stable, and unstable regions.

 



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