Wrote a rough draft of problem 5

1 solve problem 1

.gitignore

@@ -30,3 +30,9 @@
 *.exe
 *.out
 *.app
+
+# Latex
+*.aux
+*.log
+*.out
+*.bib

latex/assignment_1.tex

@@ -74,8 +74,8 @@
 
 \begin{document}
 
-\title{Title of the document}      % self-explanatory
-\author{Cory Balaton \& Janita}          % self-explanatory
+\title{Project 1}      % self-explanatory
+\author{Cory Balaton \& Janita Willumsen}          % self-explanatory
 \date{\today}                             % self-explanatory
 \noaffiliation                            % ignore this, but keep it.
 
@@ -87,6 +87,38 @@
 \section*{Problem 1}
 
 % Do the double integral
+\begin{align*}
+    u(x) &= \int \int \frac{d^2 u}{dx^2} dx^2\\
+         &= \int \int -100 e^{-10x} dx^2 \\
+         &= \int \frac{-100 e^{-10x}}{-10} + c_1 dx \\
+         &= \int 10 e^{-10x} + c_1 dx \\
+         &= \frac{10 e^{-10x}}{-10} + c_1 x + c_2 \\
+         &= -e^{-10x} + c_1 x + c_2
+\end{align*}
+
+Using the boundary conditions, we can find $c_1$ and $c_2$ as shown below:
+
+\begin{align*}
+    u(0) &= 0 \\
+    -e^{-10 \cdot 0} + c_1 \cdot 0 + c_2 &= 0 \\
+    -1 + c_2 &= 0 \\
+    c_2 &= 1
+\end{align*}
+
+\begin{align*}
+    u(1) &= 0 \\
+    -e^{-10 \cdot 1} + c_1 \cdot 1 + c_2 &= 0 \\
+    -e^{-10} + c_1 + c_2 &= 0 \\
+    c_1 &= e^{-10} - c_2\\
+    c_1 &= e^{-10} - 1\\
+\end{align*}
+
+Using the values that we found for $c_1$ and $c_2$, we get
+
+\begin{align*}
+    u(x) &= -e^{-10x} + (e^{-10} - 1) x + 1 \\
+         &= 1 - (1 - e^{-10}) - e^{-10x}
+\end{align*}
 
 \section*{Problem 2}
 
@@ -104,11 +136,19 @@
 
 \subsection*{a)}
 
+% Phrase it better
+$n = m - 2$, since $\textbf{A}$ is used to solve for all of the points in 
+between the end-points $0$ and $1$. For the complete solution, we need to add
+$u(0)$ and $u(1)$.
+
 \subsection*{b)}
 
-\section{Problem 6}
+When solving for $\vec{v}$, we find the approximate solutions for $u(x)$
+that are in between the end-points, but not the end-points themselves.
 
-\subsection{a)}
+\section*{Problem 6}
+
+\subsection*{a)}
 
 % Use Gaussian elimination, and then use backwards substitution to solve the equation
 
@@ -116,9 +156,7 @@
 
 % Figure it out
 
-\section*{Problem 7}
-
-% Link to relevant files on gh and possibly add some comments
+% Linelevant files on gh and possibly add some comments
 
 \section*{Problem 8}
 
@@ -126,7 +164,7 @@
 
 \section*{Problem 9}
 
-% Show the algorithm, then calculate FLOPs, then link to relevant files
+% Shon*{Proalgorithm, then calculate FLOPs, then link to relevant files
 
 \section*{Problem 10}