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z?{_5%+^Y9wlIQ9$#b+QDV2ZGi;ivnP4*7Db`CCI|H+VabS(62^pM2FLNdaT{|iDLikkoc literal 0 HcmV?d00001 diff --git a/latex/assignment_1.tex b/latex/assignment_1.tex index 1a875ae..a03bc4c 100644 --- a/latex/assignment_1.tex +++ b/latex/assignment_1.tex @@ -74,7 +74,7 @@ \begin{document} -\title{Title of the document} % self-explanatory +\title{Project 1} % self-explanatory \author{Cory Balaton \& Janita} % 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} @@ -106,9 +138,9 @@ \subsection*{b)} -\section{Problem 6} +\section*{Problem 6} -\subsection{a)} +\subsection*{a)} % Use Gaussian elimination, and then use backwards substitution to solve the equation