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Discussion of Runge-Kutta for Systems The methods discussed in previous tabs (refer to menu)  "ODE - Euler Method" and   "ODE - 2nd-Order Runge-Kutta" and   "ODE - 4th-Order Runge-Kutta" apply to only a single first-order ordinary differential equation as described in these tabs.

However, most problems in engineering governed by differential equations are either high-order equations or coupled differential equation systems.

A high-order differential equation can always be transformed into a coupled first-order system of equations. The trick is to expand higher-order derivatives into a series of first-order equations.

A very common example described in technical literature applies to model a spring-mass system with damping, which describes the use and calculation of second-order differential equations. We followed very closely these techniques for our own implementation.

 m d2x / d t2 = -kx - b dx/dt

where k is the spring constant and b is the damping coefficient. Since the velocity:

 v = dx/dt

the equation of motion for a spring-mass system can be rewritten in terms of two first-order 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 higher-order equations. This means that they must be expanded into a series of first-order differential equations before they can be solved using numerical methods.

The next tab shows extending the fourth-order Runge-Kutta method discussed previously to a system of ordinary differential equations.

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