Preface

Given a single ordinary differential equation, one method of finding numerical solutions entails transferring it into an equivalent system of differential equations of the first order. When a single differential equation has an isolated highest derivative, it is always possible to transfer the differential equation into an equivalent system of differential equations of the first order. This can be achieved by denoting sequential derivatives of the unknown variable with a new dependent variable with the exception of the last derivative, which is used to incorporate the given single differential equation. Other options are also possible for such a conversion, which will be clear from the examples.

The word "equivalent" indicates that both a single differential equation and the corresponding system of equations have the same set of solutions (so no solution is lost or added). The opposite is not always true, and some systems of ordinary differential equations cannot be reduced to an equivalent single differential equation of higher order. A key example highlighting this is the pendulum equation, described later in this section, which does not use the derivative operator, and where a single second order equation is transformed into four first order equations representing an equivalent system. On the other hand, the derivative operator is used in the first example, showing that the resulting system of equations is not equivalent to the original. Therefore, the key takeaway is that the only way to maintain an equivalent system is to avoid using the derivative operator because it is an unbounded operation.

There are a couple of reasons why it is convenient to transfer a single differential equation of an order higher than one to a system of differential equations of first order, such as the pendulum equation. First of all, it is simpler to analyze theoretically the first order vector differential equation than deal with a higher order differential equation because the latter can be naturally included into a more broader topic: systems of differential equations. Second, it is easier to use and implement a universal numerical algorithm for a first order derivative operator instead of utilizing special numerical procedures for higher order derivatives. It is more accurate to monitor errors of numerical calculations for a system of first order differential equations than for a single equation. Also, there are many situations in which we not only want to know the solution to an ODE (acronym for Ordinary Differential Equations) but also the derivative or acceleration of the solution. This information can be naturally extracted from the solution to the system of first order differential equations (with no extra work).

When treating the independent variable as a time, it is common to use Newton's notation \( \dot{y} \) for the derivative instead of Lagrange notation y' or Leibniz's notation \( \displaystyle {\text d}y/{\text d}t . \) Sometimes, we also utilize Euler's notation for the derivative operator: \( \displaystyle \texttt{D} = {\text d}/{\text d}t . \)