We have seen before how to solve some type of second order equations. In fact, we have only seen how to solve linear equations. So what happened to nonlinear equations ? Unfortunately many of real life problems are modelled by nonlinear equations. Here we will show how a second order equation may rewritten as a system. The technique developed for the system may then be used to study second order equation even if they are not linear.

Any second order differential equation is given (in the explicit form) as

Let us introduce the function

Then we have

Putting everything together we get

It is very easy to see that *y*(*t*) is solution to the second order
equation if and only if (*y*,*v*) is solution to the system. Keep in
mind, that our original problem deals with *y*(*t*). Therefore, the
phase plane of the system is not as important as if we were only
dealing with the associated system. This is another reason why we
should also pay attention to the graph of *y* versus *t*.

**Harmonic Oscillator**

The mass-spring apparatus is called the **Harmonic oscillator** and
is one of the most important models in science (specially in physics).
This model also rises in circuit theory (RLC circuits) and in physics
of particles.

A very rough description of the mass-spring apparatus is:

- the mass
*m*of the attached object to the spring; - the spring constant (which is a direct result of Hooke's law);
- the coefficient damping which associated to the
milieu where the spring-mass live. The damping force may be
proportional to the velocity vector or have a very complicated form
(not linear at all).

in the absence of an external force acting on the object. Recall that
*y*(*t*) denotes the position of the object at time *t*. We are clearly
assuming that the motion is linear (that is along a line). Before we
write down the associated system, we rewrite the equation in the
explicit form

The associated system is

**Undamped Harmonic Oscillators**

These are harmonic oscillators for which . In this case, the
differential equation reduces to

or equivalently

or

The associated system is

**Example.** Consider the harmonic oscillators

**1.**-
*m*=1, and ; **2.**-
*m*=1, and ; **3.**-
*m*=1, and .

1. | |

2. | |

3. |

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