- Display the direction field
for the differential equation
dy/dt = 2 cos
t - ty.
- The initial value problem
for which we will construct approximate solutions is
dy/dt = 2 cos
t - ty with y(0) = 2,
on the interval [0,8].
Calculate a 20-step Euler's Method approximation, and plot the results.
- Change the number of steps
in your worksheet to 40, recalculate, and replot (change the color if possible).
Repeat for 80 steps, and plot all three results together. Describe the
changes you see as the number of steps goes from 20 to 40 to 80.
- Now experiment with the
Improved Euler's Method: Obtain the Improved Euler's plot for 20, 40, and
80 steps. Again, describe the changes you see.
- Next calculate and plot a Fourth-Order
Runge-Kutta approximation with 40 steps. We may reasonably expect that this
solution is very close to the "exact" solution. Compare each of
your Euler's Method and Improved Euler's Method plots with the Runge-Kutta
plot. What can you say about the significance of numbers of steps for each
of the methods? Comment on anything that looks surprising.
- Finally, recalculate the Euler
approximation with only 10 steps, and plot the results. Describe what is happening
and why. Pay attention to the scales on your plot.
