Your Algebra Homework Can Now Be Easier Than Ever!

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
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 .


Reduction of order :

Use undetermined coefficients :


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
and in them.

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

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

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
bad that we spent all that time studying very special input signals.

One example is when : then is a solution to the
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
linear combinations of products of polynomials and exponential

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

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


Now write in polar form . Do the denominator first :

Lesson: if


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!

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:

OR is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for visitors -- if you order Algebra Helper by midnight of July 18th you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:

Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)


Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:

OR is an authorized reseller
of goods provided by Sofmath
Check out our demo!
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2024, Algebra-Answer.Com.  All rights reserved.