Merge pull request #73 from davidrpugh/hotfix-examples-notebook

Hotfix for slight notebook issues raised by @jstac.

examples/solving_initial_value_problems.ipynb

@@ -1,7 +1,7 @@
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@@ -13,9 +13,9 @@
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      "source": [
       "<center>\n",
-      "#Solving initial value problems (IVPs) in `quantecon`\n",
+      "# Solving initial value problems (IVPs) in `quantecon`\n",
       "\n",
-      "## David R. Pugh"
+      "## David R. Pugh\n"
      ]
     },
     {
@@ -48,7 +48,7 @@
       "\\textbf{y}'= \\textbf{f}(t ,\\textbf{y}) \\tag{1.1}\n",
       "\\end{equation}\n",
       "\n",
-      "where $\\textbf{f}:[t_0,\\infty) \\times \\mathbb{R}^n\\rightarrow\\mathbb{R}^n$.  In the case where $n=1$, then equation 1.1 reduces to a single ODE; when $n>1$, equation 1.1 defines a system of ODEs. ODEs are one of the most basic examples of functional equations: the solution to equation 1.1 is a function $\\textbf{y}(t): D \\subset \\mathbb{R}\\rightarrow\\mathbb{R}^n$. \n",
+      "where $\\textbf{f}:[t_0,\\infty) \\times \\mathbb{R}^n\\rightarrow\\mathbb{R}^n$.  In the case where $n=1$, then equation 1.1 reduces to a single ODE; when $n>1$, equation 1.1 defines a system of ODEs. ODEs are one of the most basic examples of functional equations: a solution to equation 1.1 is a function $\\textbf{y}(t): D \\subset \\mathbb{R}\\rightarrow\\mathbb{R}^n$. There are potentially an infinite number of solutions to the ODE defined in equation 1.1. In order to reduce the number of potentially solutions, we need to impose a bit more structure on the problem. \n",
       "\n",
       "## Initial value problems (IVPs)\n",
       "An [initial value problem (IVP)](http://en.wikipedia.org/wiki/Initial_value_problem) has the form\n",
@@ -65,7 +65,7 @@
       "[Finite-difference methods](http://en.wikipedia.org/wiki/Finite_difference_method) are perhaps the most commonly used class of numerical methods for approximating solutions to IVPs. The basic idea behind all finite-difference  methods is to construct a difference equation \n",
       "\n",
       "\\begin{equation}\n",
-      "\\textbf{y}(t_i)'= \\textbf{f}(t_i ,\\textbf{y}(t_i)) \\tag{1.3}\n",
+      "\\frac{\\textbf{y}(t_i + h) - \\textbf{y}_i}{h} \\approx \\textbf{y}'(t_i) = \\textbf{f}(t_i ,\\textbf{y}(t_i)) \\tag{1.3}\n",
       "\\end{equation}\n",
       "\n",
       "which is \"similar\" to the differential equation at some grid of values $t_0 < \\dots < t_N$. Finite-difference methods then \"solve\" the original differential equation by finding for each $n=0,\\dots,N$ a value $\\textbf{y}_n$ that approximates the value of the solution $\\textbf{y}(t_n)$.\n",