From c45cb6676c247cd390e824e62eab350c606e6c7f Mon Sep 17 00:00:00 2001
From: Achmad Fathoni <fathoni.id@gmail.com>
Date: Fri, 7 Jan 2022 01:02:02 +0700
Subject: [PATCH] Elaborate how c_0=x_0

---
 07-Kalman-Filter-Math.ipynb | 7 +++++++
 1 file changed, 7 insertions(+)

diff --git a/07-Kalman-Filter-Math.ipynb b/07-Kalman-Filter-Math.ipynb
index e44d74f..479e35e 100644
--- a/07-Kalman-Filter-Math.ipynb
+++ b/07-Kalman-Filter-Math.ipynb
@@ -294,6 +294,13 @@
     "x = e^ce^{kt} \\\\\n",
     "x = c_0e^{kt}\\end{gathered}$$\n",
     "\n",
+    "When $t=0$, $x=x_0$. Substitute these to equation above.\n",
+    "\n",
+    "$$\\begin{gathered}x_0 = c_0e^{k(0)} \\\\\n",
+    "x_0 = c_01 \\\\\n",
+    "x_0 = c_0 \\\\\n",
+    "x = x_0e^{kt}\\end{gathered}$$\n",
+    "\n",
     "Using similar math, the solution to the first-order equation \n",
     "\n",
     "$$\\dot{\\mathbf x} = \\mathbf{Ax} ,\\, \\, \\, \\mathbf x(0) = \\mathbf x_0$$\n",