added clarity on complex solution in homogeneous

1 files changed, 10 insertions, 2 deletions

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