From ae21dc18ebcc800403e6edab36a317fb3b9278f2 Mon Sep 17 00:00:00 2001 From: Holden Rohrer Date: Tue, 14 Apr 2020 21:14:09 -0400 Subject: improved vector representation and extended nonhomogeneous expl --- execsumm/document.tex | 13 ++++++++----- 1 file changed, 8 insertions(+), 5 deletions(-) diff --git a/execsumm/document.tex b/execsumm/document.tex index 86eea45..64f12d7 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -54,12 +54,12 @@ $\displaystyle z = y{C_2R_1\over r+C_1R_1}.$ \bu $\lambda_1 = 0, v_1 = \pmatrix{1\cr0\cr0}$ -\def\num#1{\pmatrix{1\cr - -{r_#1\over C_2\rload}\cr - -{R_1r_#1\over \rload(r_#1+C_1R_1)}}} +\def\num#1{\pmatrix{\rload\cr + -{r_#1\over C_2}\cr + -{R_1r_#1\over (r_#1+C_1R_1)}}} \bu $\displaystyle\lambda_2 = {r_1C_1C_2R_1\rload}, v_2 = \num1.$ - +\vskip 10pt \bu $\displaystyle\lambda_3 = {r_2C_1C_2R_1\rload}, v_3 = \num2.$ This corresponds to a solution of the form $f = Ce^{\lambda t}v,\cdots$ @@ -74,6 +74,9 @@ $g = C_1\cos(\re(\lambda)t)\re(v) + C_2\sin(\re(\lambda)t)\im(v).$ %% DOUBLE CHE Extending to the nonhomogeneous system will take slightly different paths depending on if the system has complex roots or has real roots. But in either case, $\cos x*{\rm polynomial}+\sin x*{\rm polynomial}$ -should be a particular solution. +should be a particular solution. This would allow usage of method of +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. \bye -- cgit