Ordinary Differential Equations

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.