ODE Workbench

Systems of Differential Equations

Description Variable Value Rate
time

position
velocity

Second order differential equations, such as Newton's second law, can be rewritten as a coupled system of first order differential equations by explicitly stating that the velocity is the rate of change of position.  For example, the system of differential equations that describe a simple harmonic oscillator are shown above.  Notice that we now have extra equations for velocity and time. The time variable has a rate of unity,  dt/dt = 1.  You can vary the time scale by adjusting this rate.  A rate of unity means that the time variable will evolve at the same rate as the applet's animation clock.

The first two equations define the two variables that will be plotted on the horizontal and vertical axes, respectively.  You can change the order of the two equations to display v(t) or a phase space plot. Press the create system button after you have entered new values.