In all experiments study the phase space diagrams too!
E 1: Run cosine and try the points and the trace option.
What do the phase space projections mean?
Try different step widths.
E 2: Go back, and chose new initial conditions. Start creates the solution, which is different from the first one.
Try points an trace option.
E 3: Create a set of solutions with identical initial value for y and different ones for y´. What is the result of different y´ for the sine function?
E 4: Create a set of solutions with identical y0 and different x0. Why do you see curves that are shifted parallel?
E 5: Create a set of curves with different initial values for y and y´, including negative ones. Interpret the results by analyzing the differential equation..
E 6: First choose Exponential, then Exponential Damping. Observe the phase space diagrams. What is the difference? Change initial values and compare again.
E 7: Choose hyperbolic sine with default initial values y = 1 y´= 1.
Now choose hyperbolic cosine with default initial values y = 1 y´= 0.
Analyze the phase space diagrams.
Remarks: For the normal exponential the gradient at x = 0 is equal to the initial value of y and cannot be zero for a meaningful exponential. Gradient zero for finite y is characteristic for the hyperbolic cosine (e x + e- x)/2, gradients > 0 with initial y = 0 for the hyperbolic sine (e x- e- x)/2. Imagine both functions mirrored at the zero-ordinate for completeness.
E 8: Choose slowing oscillation and study how the dependence on x influences the periods. Edit the formula such that frequency increases and slowing decreases.
E 9: Choose increasing oscillation and edit formulas correspondingly. Compare the effect of proportional and of reciprocal dependencies on x. Try nonlinear dependencies.
E 10: Choose damped oscillation. Check if periods are constant (when clicking at a point its coordinates are shown in the lower left corner).
E 11: Choose increasing oscillation and compare the results to those of damped oscillation. Superimpose both curves and check if periods are equal.
E 12: Draw conclusions as to which consequences different terms in the differential equation have. With that in mind, construct differential equations that will show specific characteristics.