From 549ca446e19bc8d9be162139756dce870fe38b7e Mon Sep 17 00:00:00 2001 From: Holden Rohrer Date: Tue, 14 Apr 2020 19:51:48 -0400 Subject: added clarity on complex solution in homogeneous --- execsumm/document.tex | 12 ++++++++++-- 1 file changed, 10 insertions(+), 2 deletions(-) (limited to 'execsumm') diff --git a/execsumm/document.tex b/execsumm/document.tex index 631396b..95146a4 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -60,10 +60,18 @@ $\displaystyle z = y{C_2R_1\over r+C_1R_1}.$ \bu $\displaystyle\lambda_3 = {r_2\over C_1C_2R_1\rload}, v_3 = \num2.$ -This corresponds to a solution of the form $e^{\lambda t}v,\cdots.$ +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. \section{Nonhomogeneous System} -Extending to the nonhomogeneous system, +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 \bye -- cgit