Ordinary Differential Equations

Abstract

An ordinary differential equation (ODE) in its most general form reads

$$ L(x,y,y',y'', \ldots y^{(n)} ) = 0 $$

(4.1)

where y(x) is the solution function and y′ ≡ 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{x, y) or y″ = g(x,y). 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, d²x/dt² = −ω ²₀ x, may be transformed into the system of equations

$$ \frac{{dx}} {{dt}} = v; \frac{{dv}} {{dt}} = - w_0^2 x $$

(4.2)