From 88fee54ec24bfbe9e17be81c7db80d71469ddbaa Mon Sep 17 00:00:00 2001 From: Holden Rohrer Date: Tue, 14 Apr 2020 22:34:55 -0400 Subject: added possible generalization section --- execsumm/document.tex | 17 +++++++++++++++-- 1 file changed, 15 insertions(+), 2 deletions(-) diff --git a/execsumm/document.tex b/execsumm/document.tex index 64f12d7..85d2e87 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -65,9 +65,9 @@ $\displaystyle z = y{C_2R_1\over r+C_1R_1}.$ This corresponds to a solution of the form $f = Ce^{\lambda t}v,\cdots$ where $C\in {\bf C},$ $f\in {\bf R}\to{\bf R}$. If the non-trivial eigenvectors are complex, %% TRY TO PROVE THIS!! -\def\re{{\rm Re}}\def\im{{\rm Im}} their exponential solutions form, in the reals, -$g = C_1\cos(\re(\lambda)t)\re(v) + C_2\sin(\re(\lambda)t)\im(v).$ %% DOUBLE CHECK. +\def\re{{\rm Re}}\def\im{{\rm Im}} +$g = C_1\cos(\re(\lambda)t)\re(v) + C_2\sin(\re(\lambda)t)\im(v).$ \section{Nonhomogeneous System} @@ -79,4 +79,17 @@ undetermined coefficients, which may be attempted, but variation of parameters will be used so that a general form for both systems with both complex and real eigenvalues may be found. +\section{Possible Generalization} + +This solution is general to any formulation of the original problem, +but gain may look different for a square or triangular wave, for +example. It is expected that these would exhibit similar behavior +to the sine wave because, because they would input similar amounts of +energy on a similar time scale, but the sensitivity to waveform type +could be investigated in the same way that this paper did, with possible +usage of the Laplace transform to handle discontinuities, but because +there is a general form for a sine wave and the output is proportional +to the input, a Fourier transform could be used to either approximate +or analytically obtain a solution for these types of waves. + \bye -- cgit