Differential equations

A differential equation is an equation with a derivative in it. For example, a first order differential equation is,

$$ a\frac{dx}{dt}+bx = c,$$

where $a$, $b$ and $c$ are constants. The solution to this differential equation is $x = \frac{c}{b} - \frac{a}{b}\exp \left(-\frac{b}{a}t\right)$. This can be verified by substituting the solution into the differential eqaution.

A second order differential equation contains a second derivative. Newton's law for a particle moving only in the $x$-direction can be written as a second order differential equation.

$$F_x(r_x,v_x,t) = m\frac{d^2r_x}{dt^2},$$

where the acceleration was written as the second derivative of the position, $a_x = \frac{d^2r_x}{dt^2}$. In general, the force can be a function of the position, the velocity, and the time. Every second order differential equation can be written as two coupled first order equations. For Newton's law this would be,

$$v_x = \frac{dr_x}{dt} \hspace{1.5cm} \frac{dv_x}{dv} = \frac{F_x(r_x,v_x,t)}{m}.$$

This is a convenient form for Newton's law in the cases when we don't know any of the quantities $r_x$, $v_x$, $a_x$, or $F_x$ as a function of the time. If the force is known, the trajectory of the particle can be calculated. Sometimes it is possible to find an analytic solution to a differential equation. Often, however, differential equations must be solved numerically. We will begin with a discussion of the numerical solutions of Newton's law as a differential equation.