added SEc III C

ode.tex

@@ -59,22 +59,22 @@ For the problems in this section, you may use Scipy's high-level interface to va
 \item Compute the lapse function $e^{2\Phi(r)}$ by solving Eq.(\ref{eq:tov3}) starting from $r = r_\star$ to $r = 0$. On top of that, plot the lapse function for a Schwarzschild black hole (see Eq.\ref{eq:lapse_schwarz}) from $r = r_s$ to $r = 2 r_\star$, where $r_s \equiv 2 G m_\star/c^2$ is the Schwarzschild radius of the star. This exterior solution should match the interior solution at $r = r_\star$.  
 \item Compute the lapse function $e^{2\Phi(r)}$ by solving Eq.(\ref{eq:tov3}) starting from $r = r_\star$ to $r = 0$. On top of that, plot the lapse function for a Schwarzschild black hole (see Eq.\ref{eq:lapse_schwarz}) from $r = r_s$ to $r = 2 r_\star$, where $r_s \equiv 2 G m_\star/c^2$ is the Schwarzschild radius of the star. This exterior solution should match the interior solution at $r = r_\star$.  
 \end{enumerate}
 \end{enumerate}
 
 
-% \subsection{Non-linear ordinary differential equations showing chaotic behavior: Lorenz equations}
-% The Lorenz equations were originally developed as a simplified mathematical model for atmospheric convection by Edward Lorenz. This was the first set of equations where deterministic chaos was observed. These coupled ordinary differential equations are 
-% \begin{eqnarray}
-% 	\frac{dx(t)}{dt} & = & \sigma [y(t)-x(t)], \nonumber \\
-% 	\frac{dy(t)}{dt} & = & x(t) [\rho - z(t)] - y(t), \nonumber \\
-% 	\frac{dz(t)}{dt} & = & x(t)y(t) - \beta z(t),
-% \end{eqnarray}
-% where $\rho, \sigma$ and $\beta$ are parameters of the system. 
-% \subsubsection{Problems:}
-% \begin{enumerate}
-% \item Solve the Lorenz system for $\rho = 28, \sigma = 10$ and $\beta = 8/3$ with the following initial conditions $x(t = 0) = y(t = 0) = z(t=0) = 1$. Plot $x(t)$, $y(t)$ and $z(t)$ for $t = 0 ... 100$. Is the solution deterministic or stochastic? 
-% \item Repeat the calculation with same parameters except for a tiny change in the initial condition for $x$: i.e., $x(t = 0) = 1 + 10^{-9}$. Plot  $x(t = 0) = y(t = 0) = z(t=0) = 1$ on top of the earlier estimate. Explain the result. 
-% \item Make a 3D plot of $x, y, z$. You should see the famous butterfly shaped structure now! 
-% \item Optional exercise: Make an animation of the above~\footnote{You can either use the matplotlib \href{http://matplotlib.org/api/animation_api.html}{animation} package or convert a number of PNG files to a gif animation using \href{http://www.imagemagick.org/Usage/anim_basics/}{ImageMagick}.}. 
-% \end{enumerate}
-% 
+\subsection{Non-linear ordinary differential equations showing chaotic behavior: Lorenz equations}
+ The Lorenz equations were originally developed as a simplified mathematical model for atmospheric convection by Edward Lorenz. This was the first set of equations where deterministic chaos was observed. These coupled ordinary differential equations are 
+ \begin{eqnarray}
+ 	\frac{dx(t)}{dt} & = & \sigma [y(t)-x(t)], \nonumber \\
+ 	\frac{dy(t)}{dt} & = & x(t) [\rho - z(t)] - y(t), \nonumber \\
+ 	\frac{dz(t)}{dt} & = & x(t)y(t) - \beta z(t),
+ \end{eqnarray}
+ where $\rho, \sigma$ and $\beta$ are parameters of the system. 
+ \subsubsection{Problems:}
+ \begin{enumerate}
+ \item Solve the Lorenz system for $\rho = 28, \sigma = 10$ and $\beta = 8/3$ with the following initial conditions $x(t = 0) = y(t = 0) = z(t=0) = 1$. Plot $x(t)$, $y(t)$ and $z(t)$ for $t = 0 ... 100$. Is the solution deterministic or stochastic? 
+ \item Repeat the calculation with same parameters except for a tiny change in the initial condition for $x$: i.e., $x(t = 0) = 1 + 10^{-9}$. Plot  $x(t = 0) = y(t = 0) = z(t=0) = 1$ on top of the earlier estimate. Explain the result. 
+ \item Make a 3D plot of $x, y, z$. You should see the famous butterfly shaped structure now! 
+ \item Optional exercise: Make an animation of the above~\footnote{You can either use the matplotlib \href{http://matplotlib.org/api/animation_api.html}{animation} package or convert a number of PNG files to a gif animation using \href{http://www.imagemagick.org/Usage/anim_basics/}{ImageMagick}.}. 
+ \end{enumerate}
+ 
 % \subsection{Stochastic ordinary differential equations: Langevin equation}
 % \subsection{Stochastic ordinary differential equations: Langevin equation}
 % 
 % 
 % The random motion of a particle in a fluid due to collisions with the molecules of the fluid, called the Brownian motion, is described by the Langevin equation: 
 % The random motion of a particle in a fluid due to collisions with the molecules of the fluid, called the Brownian motion, is described by the Langevin equation: