minor edits in 2nd-order backward euler

kyleniemeyer · Oct 2, 2023 · f9c43b2 · f9c43b2
1 parent d4c9827
commit f9c43b2
Showing 1 changed file with 5 additions and 4 deletions.
diff --git a/book/content/second-order/numerical-methods.ipynb b/book/content/second-order/numerical-methods.ipynb
@@ -338,10 +338,11 @@
     "\\begin{align}\n",
     "y_{1, i+1} &= y_{1, i} + \\Delta t \\, y_{2,i+1} \\\\\n",
     "y_{2, i+1} &= y_{2, i} + \\Delta t \\left( 10 - 6 y_{2, i+1} - 5 y_{1,i+1} \\right) \\\\\n",
-    "\\\\\n",
-    "y_{1, i+1} - \\Delta t \\, y_{2, i+1} &= y_{1,i} \\\\\n",
-    "5 \\Delta t \\, y_{1, i+1} + (1 + 6 \\Delta t) y_{2, i+1} &= y_{2,i} + 10 \\Delta t \\\\\n",
-    "\\text{or} \\quad \n",
+    "\\rightarrow y_{1, i+1} - \\Delta t \\, y_{2, i+1} &= y_{1,i} \\\\\n",
+    "5 \\Delta t \\, y_{1, i+1} + (1 + 6 \\Delta t) y_{2, i+1} &= y_{2,i} + 10 \\Delta t\n",
+    "\\end{align}\n",
+    "or, represented as a product of a coefficient matrix and a vector:\n",
+    "\\begin{align}\n",
     "\\begin{bmatrix} 1 & -\\Delta t \\\\ 5 \\Delta t & (1+6\\Delta t)\\end{bmatrix} \n",
     "\\begin{bmatrix} y_{1, i+1} \\\\ y_{2, i+1} \\end{bmatrix} &= \n",
     "\\begin{bmatrix} y_{1,i} \\\\ y_{2,i} + 10 \\Delta t \\\\ \\end{bmatrix} \\\\\n",