From 245cb2499d9114c4be35cb8a32d8be76f208d52a Mon Sep 17 00:00:00 2001
From: cprutean <cip.pruteanu@ed.ac.uk>
Date: Wed, 7 Aug 2024 15:24:14 +0100
Subject: [PATCH] Update file Checkpoint3.ipynb

---
 Checkpoints/Checkpoint3.ipynb | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/Checkpoints/Checkpoint3.ipynb b/Checkpoints/Checkpoint3.ipynb
index 4ac9de2..6c973dd 100644
--- a/Checkpoints/Checkpoint3.ipynb
+++ b/Checkpoints/Checkpoint3.ipynb
@@ -29,11 +29,11 @@
     "\n",
     "The solutions to this equation take the following forms:\n",
     "\n",
-    "$$ x=\\exp(âˆ’Î³t/2)[a\\cosh(pt)+b\\sinh(pt)]  when  Î³>2Ï‰_0  with  p^2=(Î³^2/4)âˆ’Ï‰^2_0 $$\n",
+    "$$ x=\\exp(âˆ’Î³t/2)[a\\cosh(pt)+b\\sinh(pt)]  \\text{when}  Î³>2Ï‰_0  \\text{with}  p^2=(Î³^2/4)âˆ’Ï‰^2_0 $$\n",
     "\n",
-    "$$ x=\\exp(âˆ’Î³t/2)[a+bt]  when  Î³=2Ï‰_0 $$\n",
+    "$$ x=\\exp(âˆ’Î³t/2)[a+bt]  \\text{when}  Î³=2Ï‰_0 $$\n",
     "\n",
-    "$$ x=\\exp(âˆ’Î³t/2)[a\\cos(Ï‰t)+b\\sin(Ï‰t)] when  Î³<2Ï‰_0  with Ï‰^2=Ï‰^2_0âˆ’(Î³^2/4)$$\n",
+    "$$ x=\\exp(âˆ’Î³t/2)[a\\cos(Ï‰t)+b\\sin(Ï‰t)] \\text{when}  Î³<2Ï‰_0  \\text{with} Ï‰^2=Ï‰^2_0âˆ’(Î³^2/4)$$\n",
     "\n",
     "where these three conditions are known as over damped, critically damped and under damped respectively.\n",
     "\n",
@@ -41,11 +41,11 @@
     "\n",
     "With the initial conditions that x=1 and $\\dot{x}$ =0 at t=0 the above constants, after some manipulation, become,\n",
     "\n",
-    "a=1 b=$\\gamma$/2p when $\\gamma$>2Ï‰$_0$\n",
+    "a=1 $b=\\gamma/2p$ when $\\gamma$>2Ï‰$_0$\n",
     "\n",
-    "a=1 b=$\\gamma$/2 when $\\gamma$=2Ï‰$_0$\n",
+    "a=1 $b=\\gamma/2$ when $\\gamma$=2Ï‰$_0$\n",
     "\n",
-    "a=1 b=$\\gamma$/2Ï‰ when $\\gamma$<2Ï‰$_0$\n",
+    "a=1 $b=\\gamma/2Ï‰$ when $\\gamma$<2Ï‰$_0$\n",
     "\n",
     "### Task\n",
     "\n",
-- 
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