126 \section{ODEs}

   127 

   128 \begin{frame}[fragile]

   129 \frametitle{ODE Integration}

   130 We shall use the simple ODE of a simple pendulum. 

   131 \begin{equation*}

   132 \ddot{\theta} = -\frac{g}{L}sin(\theta)

   133 \end{equation*}

   134 \begin{itemize}

   135 \item This equation can be written as a system of two first order ODEs

   136 \end{itemize}

   137 \begin{align}

   138 \dot{\theta} &= \omega \\

   139 \dot{\omega} &= -\frac{g}{L}sin(\theta) \\

   140  \text{At}\ t &= 0 : \nonumber \\

   141  \theta = \theta_0\quad & \&\quad  \omega = 0 \nonumber

   142 \end{align}

   143 \end{frame}

   144 

   145 \begin{frame}[fragile]

   146 \frametitle{Solving ODEs using SciPy}

   147 \begin{itemize}

   148 \item We use the \typ{odeint} function from scipy to do the integration

   149 \item Define a function as below

   150 \end{itemize}

   151 \begin{lstlisting}

   152 In []: def pend_int(unknown, t, p):

   153   ....     theta, omega = unknown

   154   ....     g, L = p

   155   ....     f=[omega, -(g/L)*sin(theta)]

   156   ....     return f

   157   ....

   158 \end{lstlisting}

   159 \end{frame}

   160 

   161 \begin{frame}[fragile]

   162 \frametitle{Solving ODEs using SciPy \ldots}

   163 \begin{itemize}

   164 \item \typ{t} is the time variable \\ 

   165 \item \typ{p} has the constants \\

   166 \item \typ{initial} has the initial values

   167 \end{itemize}

   168 \begin{lstlisting}

   169 In []: t = linspace(0, 10, 101)

   170 In []: p=(-9.81, 0.2)

   171 In []: initial = [10*2*pi/360, 0]

   172 \end{lstlisting}

   173 \end{frame}

   174 

   175 \begin{frame}[fragile]

   176 \frametitle{Solving ODEs using SciPy \ldots}

   177 \begin{small}

   178   \typ{In []: from scipy.integrate import odeint}

   179 \end{small}

   180 \begin{lstlisting}

   181 In []: pend_sol = odeint(pend_int, 

   182                          initial,t, 

   183                          args=(p,))

   184 \end{lstlisting}

   185 \end{frame}

   186 

   187 \section{Finding Roots}