eth-janishutz/eth-summaries
1
0
Fork 0
mirror of https://github.com/janishutz/eth-summaries.git synced 2026-03-14 10:50:05 +01:00
Code Issues Packages Projects Releases Wiki Activity

[Analysis] minor fix

Browse Source
This commit is contained in:
RobinB27
2026-01-27 14:46:39 +01:00
parent 36461c8082
commit 6b14b98145
2 changed files with 4 additions and 3 deletions
Download Patch File Download Diff File Expand all files Collapse all files

BIN
semester3/analysis-ii/cheat-sheet-rb/main.pdf
View File

Binary file not shown.

7
semester3/analysis-ii/cheat-sheet-rb/parts/03_diffeq_sol.tex
View File

@@ -67,10 +67,11 @@ $$
$$ $$
\subtext{$v_1,\ldots,v_k$ are the Multiplicities of $\alpha_1,\ldots,\alpha_k$} \subtext{$v_1,\ldots,v_k$ are the Multiplicities of $\alpha_1,\ldots,\alpha_k$}
\remark If $\alpha_j = \beta + \gamma i \in \C$ is a root, $\bar{\alpha_j} = \beta - \gamma i$ is too.\\ \remark If $\alpha_j = \beta + \gamma i \in \C$ is a root, $\bar{\alpha_j} = \beta - \gamma i$ is too.
To get a real-valued solution, apply:
\smalltext{If only solutions in $\R$ are considered, this can be rewritten:}
$$ $$
e^{\alpha_j x} + e^{\alpha_i x} = e^{\beta x}\left( \cos(\gamma x) + \sin(\gamma x) \right) c_1e^{\alpha_j x} + c_2e^{\alpha_i x} = e^{\beta x}\left( c_1\cos(\gamma x) + c_2\sin(\gamma x) \right)
$$ $$
\begin{subbox}{Explicit Homogeneous Solution} \begin{subbox}{Explicit Homogeneous Solution}