# Ordinary Differential Equations

• Franz J. Vesely
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