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-* Solving ODEs
-*** Outline
-***** Introduction
-******* What are we going to do?
-******* How are we going to do?
-******* Arsenal Required
-********* working knowledge of arrays
-********* working knowledge of functions
-*** Script
-    Welcome friends. 
-    
-    In this tutorial we shall look at solving Ordinary Differential Equations,
-    ODE henceforth using odeint in Python. In this tutorial we shall be using
-    the concepts of arrays, functions and lists which we have covered in 
-    previous tutorials.
-
-    Let's consider the classic problem of the spread of an epidemic in a
-    population.
-    This is given by the ordinary differential equation dy/dt = ky(L-y) 
-    where L is the total population and k is an arbitrary constant. For our
-    problem Let us use L=250000, k=0.00003.
-    Let the boundary condition be y(0)=250.
-
-    Let's now fire up IPython by typing ipython -pylab interpreter.    
-    
-    As we saw in one of the earlier sessions, sometimes we will need more than 
-    pylab to get our job done. For solving 'ordinary differential equations'
-    we shall have to import 'odeint' function, which is a part of the SciPy package.
-    So we run the magic command:
-
-    In []: from scipy.integrate import odeint
-
-    The details regarding `import' shall be covered in a subsequent tutorial.
-
-    We can represent the given ODE as a Python function.
-    This function takes the dependent variable y and the independent variable t
-    as arguments and returns the ODE.
-
-    Let us now define our function.
-
-    In []: def epid(y, t):
-      ....     k = 0.00003
-      ....     L = 250000
-      ....     return k*y*(L-y)
-
-    Independent variable t can be assigned the values in the interval of
-    0 and 12 with 61 points using linspace:
-
-    In []: t = linspace(0, 12, 61)
-
-    Now obtaining the solution of the ode we defined, is as simple as
-    calling the Python's odeint function which we just imported:
-    
-    In []: y = odeint(epid, 250, t)
-
-    We can plot the the values of y against t to get a graphical picture our ODE.
-
-    plot(y, t)
-    Lets now close this plot and move on to solving ordinary differential equation of 
-    second order.
-    Here we shall take the example ODEs of a simple pendulum.
-
-    The equations can be written as a system of two first order ODEs
-
-    d(theta)/dt = omega
-    
-    and
-
-    d(omega)/dt = - g/L sin(theta)
-
-    Let us define the boundary conditions as: at t = 0, 
-    theta = theta naught = 10 degrees and 
-    omega = 0
-
-    Let us first define our system of equations as a Python function, pend_int.
-    As in the earlier case of single ODE we shall use odeint function of Python
-    to solve this system of equations by passing pend_int to odeint.
-
-    pend_int is defined as shown:
-
-    In []: def pend_int(initial, t):
-      ....     theta = initial[0]
-      ....     omega = initial[1]
-      ....     g = 9.81
-      ....     L = 0.2
-      ....     f=[omega, -(g/L)*sin(theta)]
-      ....     return f
-      ....
-
-    It takes two arguments. The first argument itself containing two
-    dependent variables in the system, theta and omega.
-    The second argument is the independent variable t.
-
-    In the function we assign the first and second values of the
-    initial argument to theta and omega respectively.
-    Acceleration due to gravity, as we know is 9.81 meter per second sqaure.
-    Let the length of the the pendulum be 0.2 meter.
-
-    We create a list, f, of two equations which corresponds to our two ODEs,
-    that is d(theta)/dt = omega and d(omega)/dt = - g/L sin(theta).
-    We return this list of equations f.
-
-    Now we can create a set of values for our time variable t over which we need
-    to integrate our system of ODEs. Let us say,
-
-    In []: t = linspace(0, 20, 101)
-
-    We shall assign the boundary conditions to the variable initial.
-
-    In []: initial = [10*2*pi/360, 0]
-
-    Now solving this system is just a matter of calling the odeint function with
-    the correct arguments.
-
-    In []: pend_sol = odeint(pend_int, initial,t)
-
-    In []: plot(pend_sol)
-    This gives us 2 plots. The green plot is omega vs t and the blue is theta vs t.
-
-    Thus we come to the end of this tutorial. In this tutorial we have learnt how to solve ordinary
-    differential equations of first and second order.
-
-*** Notes