Our first example is the classical case of simple harmonic oscillations, namely a block sliding on a horizontal frictionless surface. The block is tied to a wall with a spring. If the spring is not compressed or stretched too far, the force on the block at a given position $x$ is $$F=-kx.$$ The negative sign means that the force acts to restore the object to an equilibrium position. Newton's equation of motion for this idealized system is then $$m\frac{d^2x}{dt^2}=-kx,$$ or we could rephrase it as $$\frac{d^2x}{dt^2}=-\frac{k}{m}x=-\omega_0^2x, \tag{43}$$ with the angular frequency $\omega_0^2=k/m$.

The above differential equation has the advantage that it can be solved analytically with solutions on the form $$x(t)=Acos(\omega_0t+\nu),$$ where $A$ is the amplitude and $\nu$ the phase constant. This provides in turn an important test for the numerical solution and the development of a program for more complicated cases which cannot be solved analytically.