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By Giuseppe Tenti and annotated by Douglas Wilhelm Harder.

The next three lectures will examine four 2nd-order initial-value
problems with constant coefficients including:

- Mechanical Oscillator Model
- Electrical Oscillator Model
- Free Oscillations
- Forced Oscillations
- The Schrödinger Equation
- The Particle in a Box

# The Mechanical Oscillator Model

Consider the mass attached to two equal strength springs
shown in Figure 1. The two springs
(elastic objects storing mechanical energy) are anchored to two walls
and the mass is sitting on a surface with resistance which will
oppose motion. When the mass is at position ,
the force by the two springs is equal and therefore the mass is at
equilibrium.

Figure 1. A mass at rest between two springs.

The force on the mass when it is at rest is 0, but if there
is a non-zero force on the mass, the force will cause an acceleration
due to Newton's law: . Because
acceleration is defined as the second derivative of position with
respect to time, we may write .

Suppose now that the mass is displaced to a distance
from the equilibrium point, as is shown in
Figure 2.

Figure 2. The mass in Figure 1 displaced to a distance and the resulting force.

At this point, the springs cause a force on the mass
attempting to force the mass back to equilibrium. The size of
the force is proportional to the stretch of the spring, and therefore
we may write:

where is the *spring constant*. This is called Hooke's law. If these were the only forces, our differential equation would
therefore be

where both and
. By rewriting this as

we already know that the general solution must be of the form

.

This says that, if the mass is ever displaced and released, it will oscillate forever.

In reality, however, there is a friction between the mass and the surface, as is shown in Figure 3,
and once the mass begins to move, the frictional force moves to oppose that movement.

Figure 3. The friction opposing the movement of the mass.

We can therefore add this mass to to our equation:

or

.

Note: both of these forces are *phenomenological* in the sense that they are
local linear approximations based on experimental data. If a spring is stretched too far, that is, it is pushed to
its elastic limit, it will fail, and neither spring
can be compressed beyond the wall it is attached to.

Finally, let us assume that there is a external force independent of the springs which is
also placing a force on the mass. This is a force which may vary with time, and therefore
we will represent it by a function as shown in Figure 4.

Figure 4. A force external to the springs and friction.

This gives us our final differential equation

.

Equation 1

## Standard Form

Because the differential equation in Equation 1 is so common, engineers will
generally rewrite this differential equation in the form

where

- is the
*natural frequency* of the oscillator,
- is the
*damping parameter*, and
- is the
*reduced* external force.

For this example of an oscillating mass, we have that:

- is the
*natural frequency* of the oscillator,
- is the
*damping parameter*, and
- is the
*reduced* external force.

This differential equation is a linear 2nd-order inhomogeneous differential equation with constant coefficients.
If the reduced external force is zero, the differential equation is homogeneous.

Science and engineering use the following terminology:

The differential equation describes
a *forced oscillator* and is the *forcing function*.

The differential equation describes
a *free oscillator* or an *unforced oscillator*.

# The Electrical Oscillator Model

As you have seen in your circuit theory courses, a circuit containing a capacitor and an inductor have
two energy-storage elements that cannot replaced by a single storage element. (You will recall that resistors, capacitors, and
inductors in parallel or series may normally be replaced by a single element of the same kind while two such
elements in series may be switched.) Such a circuit is called a linear second-order circuit as it is described by
a linear second-order differential equation.

While there are arbitrarily many different circuits, we will focus on the simple RLC circuit
shown in Figure 5.

Figure 5. An RLC circuit with a switch.

If there is no charge on the capacitor and no current passing through the inductor, the
circuit is said to be *quiescent*. At a moment in time, usually at time
, the switch is closed and the current begins to flow
through the wire—the circuit is now active.

To determine the activity in the circuit, it is necessary to recall that the voltage drop across

- the inductor is ,
- the resistor is , and
- the capacitor is .

Therefore, according to Kirchoff's Voltage Law (KVL or the *principle of conservation of charge*) we have

If our goal is to find the current, we may differentiate the equation:

or, if we are interested in finding the charge, we can substitute to
get

.

Both of these differential equations are mathematically equivalent. The derivative of a solution to
the first differential equation is a solution to the second, but also, both have the same natural
frequency () and the
same damping parameter ()—only the reduced external force
( or , respectively) is different.

We will write this common mathematical structure as

and we will then only have to specify the variables
,
,
, and
as well as the necessary initial conditions to get the desired solution.

# Finding the Standard Form in General

To convert the differential equation

into the standard form, divide by to get

Now, in general for reasons of stability, the ratio . Thus,
solve and for convenience, choose the positive root. We can
do this because and therefore it is simply
convenient to choose the positive frequency.

Next, solve to get
.