# Forest UCM Osc Driven

Damped Oscillators driven by an external source

An external force must be supplied to do work on a damped oscillator in an amount that is equal to or greater than the work done by the dissipative force.

An external force (source) is added to the homogeneous differential equation making it inhomogenous

making it

where f(t) represents the external force (source) that depends on time divided by the objects mass.

## Differential equations in Operator form

In the previous sections we used the definition

to solve the second order linear differential equation.

Let's take this a step further with the following operator definition

then

becomes

Linear differential equations have coefficient that can constant or variable coefficients that can be transformed into constant coefficients.

Note
is a linear operator

meaning

the above is a property of differential calculus where

and

## Solving the Inhomogeneous Diff. Eq.

To solve the Inhomogeneous problem we take advantage of the linear operator such that

Since the solution of the 2nd order differential equation requires exactly two arbitrary constant,

We can formulate the solution by tacking a particular solution onto the homogeneous solution that already has two arbitrary constants.

All we need to do is find the particular solution that satisfies

and we will have a complete set of solutions.

Break the equation up into a Homogeneous solution and a Particular Solution

### Homogeneous Solution

Th Homogenous solution solved the equation

we know from the previous section that the homogenous solution has the form

where

### Case 1 : f(t) is sinusoidal

Consider the case where the driving force is a sinusoidal function