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authorHolden Rohrer <hr@hrhr.dev>2020-04-14 21:14:09 -0400
committerHolden Rohrer <hr@hrhr.dev>2020-04-14 21:14:09 -0400
commitae21dc18ebcc800403e6edab36a317fb3b9278f2 (patch)
tree066105c206eb722a433944c33b7bb54ad734154b /execsumm
parent99350f63187004d3edac355f9c1275fc6da725ea (diff)
improved vector representation and extended nonhomogeneous expl
Diffstat (limited to 'execsumm')
-rw-r--r--execsumm/document.tex13
1 files 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