Merge pull request #8 from FYS3150-G2-2023/2-solve-problem-1

Solve problem 1

latex/assignment-2.tex

@@ -89,7 +89,7 @@
     
 \textit{https://github.uio.no/FYS3150-G2-2023/Project-2}
     
-% \input{problems/problem-1}
+\input{problems/problem-1}
 
 % \input{problems/problem-2}
 

latex/problems/problem-1.tex

@@ -0,0 +1,16 @@
+\section*{Problem 1}
+We are studying the one-dimentional buckling beam, which can be described by the equation
+\begin{align*}
+    \gamma \frac{d^{2} u(x)}{dx^{2}} &= -F u(x) &  \rightarrow & & \frac{d^{2} u(x)}{dx^{2}} &= - \frac{F}{\gamma} u(x) \\
+\end{align*}
+where $\gamma$ is a constant determined by the material of the beam. We want to scale the equation, that is we want to scale by the x-value of the beams endpoint $x=L$. 
+Scaling will result in a dimensionless variable $\hat{x} = \frac{1}{L}$. 
+%
+\begin{align*}
+        \frac{d^{2}}{dx^{2}} &= \frac{d}{dx} \frac{d}{dx} = \bigg( \frac{d\hat{x}}{dx} \frac{d}{d\hat{x}} \bigg) \bigg( \frac{d\hat{x}}{dx} \frac{d}{d\hat{x}} \bigg) & \text{where we have used } \frac{d\hat{x}}{dx} \frac{d}{d\hat{x}} = \frac{d}{dx} \frac{d\hat{x}}{d\hat{x}} \\
+        &= \bigg( \frac{1}{L} \frac{d}{d\hat{x}} \bigg) \bigg( \frac{1}{L} \frac{d}{d\hat{x}} \bigg) = \frac{1}{L^{2}} \frac{d}{d\hat{x}^{2}} & \text{where } \hat{x} \equiv \frac{x}{L} \text{ and } \frac{d\hat{x}}{dx} = \frac{1}{L} \\
+\end{align*}
+Now we insert the expression into the original equation
+\begin{align*}
+    \frac{d u(\hat{x})}{d\hat{x}^{2}} &= - \frac{F L^{2}}{\gamma} u(\hat{x}) \\
+\end{align*}

latex/problems/problem-2.tex

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+\section*{Problem 2}
+

latex/problems/problem-3.tex

@@ -0,0 +1 @@
+\section*{Problem 3}

latex/problems/problem-4.tex

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+\section*{Problem 4}

latex/problems/problem-5.tex

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+\section*{Problem 5}

latex/problems/problem-6.tex

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+\section*{Problem 6}