The methods discussed in previous tabs (refer to menu) "ODE  Euler Method" and
"ODE  2ndOrder RungeKutta" and
"ODE  4thOrder RungeKutta" apply
to only a single firstorder ordinary differential equation as described in these tabs.
However, most problems in
engineering governed by differential equations are either highorder equations or coupled differential equation
systems.
A highorder differential equation can always be transformed into a coupled firstorder system of equations. The trick is
to expand higherorder derivatives into a series of firstorder equations.
A very common example described in technical literature
applies to model a springmass system with damping, which describes the use and calculation of secondorder differential equations.
We followed very closely these techniques for our own implementation.
m d^{2}x / d t^{2} = kx  b dx/dt 

where k is the spring constant and b is the damping coefficient. Since the velocity:
the equation of motion
for a springmass system can be rewritten in terms of two firstorder differential equations:
dv/dt = k/m x  b/m v
dx/dt = v 
In the above equation, the derivative of v is a function of v and x, and the derivative of x is a function
of v. Since the solution of v as a function of time depends on x and the solution of x as a function of time
depends on v, the two equations are coupled and must be solved simultaneously.
Most of the differential equations in engineering are higherorder equations. This means that they must be expanded
into a series of firstorder differential equations before they can be solved using numerical methods.
The next tab shows extending the fourthorder RungeKutta method discussed previously to a system of ordinary
differential equations.
