From 2cb23c38bf943a86505f8d51d42e7ca0d6e626c1 Mon Sep 17 00:00:00 2001
From: Benjamin Chausse <benjamin@chausse.xyz>
Date: Tue, 3 Dec 2024 15:45:49 -0500
Subject: More helpers

---
 annexe/main.tex | 15 ++++++++++++---
 1 file changed, 12 insertions(+), 3 deletions(-)

(limited to 'annexe/main.tex')

diff --git a/annexe/main.tex b/annexe/main.tex
index f42234e..15214f3 100644
--- a/annexe/main.tex
+++ b/annexe/main.tex
@@ -28,6 +28,13 @@
 \newcommand{\vlt}{V_L(t)}
 \newcommand{\vlz}{V_L(0)}
 
+\newcommand{\e}{\mathrm{e}}
+\newcommand{\al}{\alpha}
+\newcommand{\la}{\lambda}
+\newcommand{\wz}{\omega_{0}}
+\newcommand{\ws}{\omega_{0}^2}
+\newcommand{\wn}{\omega_{\mathrm{n}}}
+
 
 \DeclareSIPrefix{\micro}{%
   \text{%
@@ -62,9 +69,11 @@
 
 	\subsection{Décharge}
 	\begin{align}
-		0 & = \vlt + V_C(t) + V_R(t)                 \\
-		0 & = \vlt+\frac{1}{C}\int I_(t)\dt + R_I(t) \\
-		0 & = \ddt{2} \vlt
+		0 & = \vlt + V_C(t) + V_R(t)                                                              \\
+		0 & = \vlt+\frac{1}{C}\int I_(t)\dt + R_I(t)                                              \\
+		0 & = \ddt{2}\left[ \vlt + \frac{1}{LC}\int\int\vlt\dt^2 + \frac{R}{L}\ddt{} \vlt \right] \\
+		0 & = \ddt{2}\vlt+ \frac{R}{L}\ddt{}\vlt +\frac{1}{LC}\vlt                                \\
+		0 & = \ddt{2}+2\al\la A\e
 	\end{align}
 \end{appendix}
 
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