Plotting slope fields and using them to solve IVPs

(1) Start up Jupyter Notebook and open a new Python3 notebook.

(2) Import the resources306 module:

from resources306 import *

(3) Create a slope field plot for the differential equation

(dy)/(dt) = cosy + ysint

over the region where t runs from 0 to 5 and y runs from -1 to 5.

Here's code similar to what I showed you in class for a different DE, dy/dx = (y-x)/4:

Notes:

Use * for multiplication.
Use Shift-Enter to execute a complete command.
Use Enter to continue a command on a new line.
You will need to import the functions sin and cos from numpy: from numpy import cos, sin
To get a usable plot, you may need to adjust the last number supplied to slopefield, which is the vertical spacing of the slope line segments.

(4) Once you have obtained a satisfactory correct slope field picture, ask the TA to check it and give you a printed copy of the slope field.

(5) On the printed slope field, sketch a curve with the following properties:

it starts at t = 0, y = 0,
its slope agrees with the slope field at every point along it.

(6) From your picture, estimate as accurately as you can y(4) if the function y(t) satisfies

y(0) = 0 and (dy)/(dt) = cosy + ysint

for all t between 0 and 4. Annotate your picture to show how you got your estimate. Take a photo of your work for your own future reference.

(7) Have the TA check the correctness of your work.