The simulation calculates solutions of ordinary explicit second order differential equations.
y´ ´= d 2y/ dx = f( y, y´, x)
using the Runge−Kutta algorithm. In the left coordinate system the abscissa represents x, the ordinate y.
When opening the file you see a fat red point at x = 0, representing the initial value y0 at its abscissa x0 . You can change the initial value with the slider, more exactly and unlimited by typing a value into the number field. Two additional number fields are used to define x0 and xmax . Default values are: y0 = 1; x0 = 0; xmax = 3. You can also draw the red point to create new initial conditions of y.
With a second slider you can define the initial value for the first derivative. It corresponds to the gradient at the initial ordinate, which is symbolized by an arrow.
In the ComboBox you can chose between a number of predefined types of functions. Their formula is shown in field y´´ = ... There you can edit formulas or input any arbitrary first order explicit differential equation.
Activating start for the default equation, the differential equation of the exponential function y´´= y is evaluated. Calculation stops as x = xmax . At first you see a set of calculated points. You can choose the option trace to see an interpolated curve.
Stop stops the calculation; back leaves already calculated points and sets back to the initial conditions. Changing these now creates an additional curve at start. This way you can create sets of solutions for different initial conditions (the trace option would create jumps, which are avoided in the points option). Clear resets and clears traces, but leaves parameters unchanged. Reset leads back to default values..
After back you can change the x-resolution of calculation by slider step, and look how different resolutions influence the result.
The smaller window shows the phase space projections
y ´ = y ´( y )
y´´ = y´´ ( y )
The thick points are the last ones calculated.
The phase space diagrams very distinctively demonstrate the different character of solutions: convergence, divergence, periodic oscillation, oscillating divergence, oscillating convergence. It is independent of the initial condition.