Higher order equations in the construction of a dynamic system model

If, when building a model of a dynamical system, it turned out that his records used derivatives of a second or more degree, this means that we are dealing with higher order equations. Higher order equations lead to a system of first order equations. Consider this by example.

Example 1. The following second order differential equation is given: y '' + y ' / x + y = 0. It is necessary to reduce it to a system of first order equations.

Denote: d y / d t = z ; d ² y / d t ² = z ' . Then: z ' + z / x + y = 0. And then we come to the system of differential equations of the first order:

Then we solve this system by means already known to us, for example, by the Euler method (see Lecture 10. Numerical methods for integrating differential equations. Euler method).