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Review of Constant Coefficient Linear Equations

Big example, superposition, and Frequency Response

[1] Example.

PLEASE KNOW the solution to the homogeneous harmonic oscillator
are sinusoids of circular frequency omega !
Here, .

In the real example I drive it: .

The complex equation is .

If it weren't for the t we could try to apply ERF: ,
, though, so it doesn't apply; we do have the
resonance
response formula, which gives
so .

But there is a t there. We should then use "Variation of Parameters":
Look for solutions of the form

for u an Unknown function.



so

Reduction of order :

Use unde termined coefficients :

 so

The general solution is then the homogeneous solution.

[3] Superposition: putting special cases together .

Suppose a bank is giving I percent per year interest:



Suppose that I open TWO bank accounts and proceed to save at rates
(t)
and in them.

Is this any different than opening ONE bank account and saving at the
rate


Say the solutions with savings rates and are and .
Is a solution with savings rate ?

since differentiation respects sums (and multiplying by I does too).

In general if and
then

In fact this is true for nonconstant coefficient linear equations too.
It is the essence of linearity, and it's the most general form of the
superposition principle.

It lets you break up the input signal into constituent parts, solve for
them
separately, and then put the results back together. This is why it isn't
so
bad that we spent all that time studying very special input signals.

One example is when : then is a solution to the
homogeneous
equation, and we find again that adding such a function to a solution of
gives another solution.

Our work has shown a general result:

Theorem: If q(t) is any linear combination of products of polynomials
and exponential   functions, then all solutions to are
again
linear combinations of products of polynomials and exponential
functions.

Here we mean *complex* linear combinations and *complex* exponentials,
so for example is a possible signal
or solution.

[4] Frequency response

Polar form of a complex number:

Frequency response is about the amplitude and phase lag of a sinusoidal
(steady state) response of a system to a sinusoidal signal of some
frequency.

It is based on the fol lowing method of finding a sinusoidal system
response in "polar" (amplitude/phase lag) form:

Example:


Now write in polar form . Do the denominator first :

Lesson: if

then



Suppose now that I let the input frequency be anything:
 

So the amplitude of the sinusoidal response is

This takes value 1 at omega = 0 , and when omega is large it
falls off like . In this case, it reaches a modest
"near resonance" peak at omega = 1 .

The phase lag is

There's no particular advantage in writing out a more explicit formula
for this.

Good luck!

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