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1.1: Ordinary Diﬀerential Equations

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ODE’s are equations involving an unknown function and its derivatives, where the function depends on a single variable, e.g., the equation for a particle moving at constant velocity,

$\frac{d}{dt} x(t) = v,$

which has the well known solution $x(t) = vt+x_0.$ The unknown constant $$x_0$$ is called an integration constant, and can be determined if we know where the particle is located at time $$t=0$$. If we go to a second order equation (i.e., one containing the second derivative of the unknown function), we find more integration constants: the harmonic oscillator equation

$\frac{d^2}{dt^2} x(t) = - \omega^2 x(t)$

has as solution $x = A \cos \omega t + B \sin \omega t,$

which contains two constants.

As we can see from the simple examples, and as you well know from experience, these equations are relatively straightforward to solve in general form. We need to know only the coordinate and position at one time to fix all constants.