- Set a = 0.5, b
= 1, y(0) = 1, and y'(0) = 0, so the initial value problem
is
| y"
+ 0.5 y' + y |
= |
0, |
| y(0) |
= |
1, |
| y'(0) |
= |
0. |
Change the range on y
to [-1,1], and graph the solution. Increase a in steps of
0.5 up to 3, and describe how the solutions change.
Use the symbolic solution to explain what you see in these solution graphs.
- Set a = -0.25, so
the initial value problem now is
| y"
- 0.25 y' + y |
= |
0, |
| y(0) |
= |
1, |
| y'(0) |
= |
0. |
Remove the range on y,
and let your computer algebra system set the range. Plot the solution and
describe the graph. Now set a = -0.5, and replot. Then repeat for
a = -0.75. Describe each of these plots. Use the symbolic solution
to explain what you see in these solution graphs.
- Set a = -1. To get
a clear mental image of this graph, plot for t in [0,4],
then repeat for [0,8], [0,12], and [0,16]. Describe
the solution graph, and use the symbolic solution to explain its form.
- Now decrease a from
-1 to -3 in steps of 0.25. Vary the t-interval
as appropriate to obtain a good understanding of each graph. Describe the
graphs of these solutions. In particular, indicate which of these solution
graphs cross the t-axis more than once. Use the symbolic solution
to explain these solution graphs.
