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authorHolden Rohrer <hr@hrhr.dev>2020-04-14 19:51:48 -0400
committerHolden Rohrer <hr@hrhr.dev>2020-04-14 19:51:48 -0400
commit549ca446e19bc8d9be162139756dce870fe38b7e (patch)
tree3ec229aaee7c389fb6c6e9ec9c3d96ba1021a17c /execsumm
parent2427b2b28b2083d82ac0c5f757d4d26249332968 (diff)
added clarity on complex solution in homogeneous
Diffstat (limited to 'execsumm')
-rw-r--r--execsumm/document.tex12
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