The harmonic oscillator is the name for system that behaves like a particle attached to an ideal elastic spring, in the absence of a gravitational field.
The force to restore equilibrium for this system is proportional to the displacement of the particle away from the equilibrium position. Since this force is opposite of the displacement we have
The proportionality constant, k, is also called the spring constant. It is equal to the force required to either compress or stretch the spring over a unit distance. It is a positive valued property of the spring and has units of N m-1 or kg s-2.
Now we can understand the requirements of ideality and elasticity of the spring. The need for the spring to behave ideally means that k has the same value for any u, i.e., is independant of u. The need for the spring to be elastic means that it will be impossible to permanently deform it, so that any amount of compression or extension will not 'break' the spring and allow it to return to the initial equilibrium position.
To determine how the positions of a particle with mass m, will be determined by this force, we need to solve Newton's equation of motion for this system, i.e.
Other ways in which you may encounter the same expression are
From this last form of the differential equation we immediately see that the solution u can be multiplied by any constant, N, and still be a solution.
There are different ways to solve the differential equation
Either way the solutions for this equation is given by
and the values of the constants A, and f0 can be found from the initial conditions for t = 0,
