diff --git a/content/GA_1_5/Analysis_solution.ipynb b/content/GA_1_5/Analysis_solution.ipynb
index 399b52b8a010d1ac256dc8a4a05cdeae957c8220..29ac086181f1df98e865e4d0d5630b96be18bf2f 100644
--- a/content/GA_1_5/Analysis_solution.ipynb
+++ b/content/GA_1_5/Analysis_solution.ipynb
@@ -491,14 +491,29 @@
     "<p>\n",
     "<b>Solution:</b>   \n",
     "\n",
-    "For backward euler we need to evaluate at $x_{i-1}$ around $x_{i}$. This yields the following Taylor approximation: \n",
+    "For backward euler we need to evaluate at $x_{i-1}$ around $x_{i}$ and $x_{i-2}$ around $x_{i}$. This yields the following Taylor approximation: \n",
     "\n",
     "$$\n",
-    "f(x_{i-1})\\approx f(x_{i})+(x_{i-1}-x_i)\\frac{\\partial f(x_{i})}{\\partial x} + \\mathcal{O}(x_{i-1}-x_i)^2\n",
+    "f(x_{i-1})\\approx f(x_{i})+(x_{i-1}-x_i)\\frac{\\partial f(x_{i})}{\\partial x} +\\frac{(x_{i-1}-x_i)^2}{2}\\frac{\\partial^2 f(x_i)}{\\partial x^2}+\\mathcal{O}(x_{i-1}-x_i)^3\n",
     "\\newline\\newline\n",
-    "\\frac{\\partial f(x_{i})}{\\partial x}\\approx \\frac{f(x_{i-1})-f(x_{i})}{x_{i-1}-x_i} + \\mathcal{O}(x_{i-1}-x_i)^2\n",
+    "f(x_{i-2})\\approx f(x_{i})+(x_{i-2}-x_i)\\frac{\\partial f(x_{i})}{\\partial x} +\\frac{(x_{i-2}-x_i)^2}{2}\\frac{\\partial^2 f(x_i)}{\\partial x^2}+\\mathcal{O}(x_{i-2}-x_i)^3\n",
     "\\newline\\newline\n",
-    "\\frac{\\partial f(x_{i})}{\\partial x}\\approx \\frac{f(x_{i})-f(x_{i-1})}{x_i-x_{i-1}} + \\mathcal{O}(x_{i-1}-x_i)^2\n",
+    "$$\n",
+    "we set $\\Delta x = x_i - x_{i-1}$, which also means: $2\\Delta x = x_i - x_{i-2}$\n",
+    "$$\n",
+    "f(x_{i-1})\\approx f(x_{i})-\\Delta x\\frac{\\partial f(x_{i})}{\\partial x} +\\frac{\\Delta x^2}{2}\\frac{\\partial^2 f(x_i)}{\\partial x^2}+\\mathcal{O}(\\Delta x)^3\n",
+    "\\newline\\newline\n",
+    "f(x_{i-2})\\approx f(x_{i})-2\\Delta x\\frac{\\partial f(x_{i})}{\\partial x} +\\frac{4\\Delta x^2}{2}\\frac{\\partial^2 f(x_i)}{\\partial x^2}+\\mathcal{O}(\\Delta x)^3\n",
+    "\\newline\\newline\n",
+    "$$\n",
+    "We multiply the first expression by 4 and subtract the second expression:\n",
+    "\n",
+    "$$\n",
+    "4f(x_{i-1})-f(x_{i-2})\\approx (4-2)f(x_{i})-(4-2)\\Delta x\\frac{\\partial f(x_{i})}{\\partial x} +(4-4)\\frac{\\Delta x^2}{2}\\frac{\\partial^2 f(x_i)}{\\partial x^2}+\\mathcal{O}(\\Delta x)^3\n",
+    "\\newline\\newline\n",
+    "3f(x_i)-4f(x_{i-1})+f(x_{i-2}) \\approx \\frac{\\partial f(x_{i})}{\\partial x} \n",
+    "\\newline\n",
+    "\\frac{3f(x_i)-4f(x_{i-1})+f(x_{i-2})}{2\\Delta x} \\approx \\frac{\\partial f(x_{i})}{\\partial x}+\\mathcal{O}(\\Delta x)^2\n",
     "$$\n",
     "\n",
     "</p>\n",