# Ordinary Differential Equations

Chapter

## Abstract

An ordinary differential equation (ODE) in its most general form reads
$$L\left( {x,y,y',y'', \ldots {y^{\left( n \right)}}} \right) = 0$$
(4.1)
where y(x) is the solution function and $$y' \equiv dy/dx$$ etc. Most differential equations that are important in physics are of first or second order, which means that they contain no higher derivatives such as $$y'''$$ or the like. As a rule one may rewrite them in explicit form, $$y' = f\left( {x,y} \right)ory'' = g\left( {x,y} \right)$$. Sometimes it is profitable to reformulate a given second-order DE as a system of two coupled first-order DEs. Thus, the equation of motion for the harmonic oscillator, $${d^2}x/d{t^2} = - \omega _0^2x$$ ,may be transformed (introducing the auxiliary function υ(t)) into the system
$$\frac{{dx}}{{dt}} = \upsilon ;\frac{{d\upsilon }}{{dt}} = - \omega _0^2x$$
(4.2)
Another way of writing this is
$$\frac{{dy}}{{dt}} = L\cdot y,wherey \equiv \left( {\begin{array}{*{20}{c}} x \\ \upsilon \\ \end{array} } \right)andL = \left( {\begin{array}{*{20}{c}} 0 & 1 \\ { - \omega _{0}^{2}} & 0 \\ \end{array} } \right)$$
(4.3)
As we can see, у and d у/dt occur only to first power: we are dealing with a linear differential equation.

## Keywords

Boundary Value Problem Harmonic Oscillator Initial Value Problem Kepler Problem Local Truncation Error
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.