\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*}
