Using Row-Echelon Reduction With Differential Operators As Elements

Hello. If this question belongs in another forum, I apologize.

When solving 'n-1' homogeneous equations in 'n' unknowns, is it permissible to use

differential operators in the matrix elements?

For example, when trying to solve :

i = i1 + i2

i = Cv' - Cx'

i1 = x/R

i2 = i3 + i4

i2 = Cx' - Cy'

i3 = y/R

i4 = Cy' - Cu'

i4 = u/R

The variables i, i1, i2, i3 and i4 are instantaneous currents {i.e., i = i(t)}.

The variables x, y, u and v are instantaneous voltages.

The prime marks indicate first derivatives.

I was thinking about putting these equations in row-echelon form, and then

performing reductions. For the primed variables, I was thinking about using

the operator D = d/dt.

Can I do this, being aware that I treat the operator algebraically in the columns,

but use it operationally row-wise?

Thanks!!

Problem Definition for Example Given

This problem comes from Paul J. Nahin's book

"An Imaginary Tale (The Story of {radical -1})", Princeton University Press, 1998,

on page 139. This example is a circuit solution for a phase-shift oscillator.

The author says :

"With a total of eight equations in nine variables we can solve for the ratio of any

two, and the particular ration of interest for our circuit is u/v. One way to do this

is to first manipulate the above equations to eliminate all the variables except

for u and v.

This is not difficult to do, but it is rather detailed, and so I will just give you the

answer and encourage you to verify it :

v''' = u''' + [6/RC]u'' + [5/{(RC)^2}]u' + [1/{(RC)^3}]u ."

I was hoping in this instance I could use row-echelon form, reducing to the two

variables u and v. In this situation, will this method work?

If not, would you please suggest a path to follow.

Thanks!

Clearing up semantic ambiguity

Hi Adrian,

I probably should have written more than that single sentence, which is semantically very vague - sorry about that!

When I was multiplying by D across rows, the elements in each column were simple polynomials in D.

For example : suppose a (zero-augmented) 2x3 echelon matrix,

with left-to-right column headings of 'y', 'v', and 'u', containing two rows

{1, -[(RC)^2](D^2), [4 + 6RCD + [(RC)^2](D^2)]} and

{CD, 0, [-1/R - CD]} .

[D is defined to be "d/dt"; R and C are constants.]

{The two rows represent the equations :

0 = y - [(RC)^2]v'' + 4u + 6RCu' + [(RC)^2]u'' and

0 = Cy' - u/R - Cu' }

Algebraically multiplying the first row by (-CD) and

adding the result to the second row yields :

{0, (R^2)(C^3)(D^3), [-1/R - 5CD - 6R(C^2)(D^2) - (R^2)(C^3)(D^3)]}

which represents :

0 = (R^2)(C^3)v''' - u/R -5Cu' - 6R(C^2)u'' - (R^2)(C^3)u'''

(which gives the answer the author mentioned above wanted).

This is probably so elementary a thing that I overcomplicated it with my lousy ability in English composition.