E 1: Choose the number of intervals as n = 5. Visually compare the quality of the three numerical approximations. Use the right window with its zoomed scale for that purpose, too.
E 2: Change the number of intervals n and compare qualitatively how the three approximations approach the analytic solution.
E 3: Make notes of the relative mistakes of the three methods with increasing n and draw a graph of their dependences. Compare this graph with a straight line for Euler and with a second grade parabola for Heun (with Runge−Kutta the deviations will be too small to clearly recognize the graph as a fourth grade parabola).
E4: Change the initial value. What is the influence on the relative mistake? Why?
E 5: Choose (initial value) = 1 and n = 3; The width of the interval now is 1. Use the cyan lines to construct the first and second steps for Euler and Heun (see drawing in book text). The lines can be shifted and rotated with the mouse.
E 6: Choose (initial value) = 1 and n = 2; now Runge−Kutta will show a distinct deviation for the most distant point. Try to reconstruct its Algorithm with the cyan lines (most probably you will need paper and pencil in addition). Remember that the first derivative for the next point is equal to the exponential at its analytic ordinate for the exponential function (and only for it!).
E 7: Open the file from the EJS console and study the code at page Initialization. Use other derivatives to solve different differential equations (for the exponential the derivative was simply y). The correct places are marked in red.