\documentclass[12pt]{article}

\title{Solving Equations \& ODEs}

\author{FOSSEE}

\begin{document}

\date{}

\vspace{-1in}

\begin{center}

\LARGE{Solving Equations \& ODEs}\\

\large{FOSSEE}

\end{center}

\section{Solving linear equations}

\begin{verbatim}

    In []: A = array([[3,2,-1],

                      [2,-2,4],                   

                      [-1, 0.5, -1]])

    In []: b = array([[1], [-2], [0]])

    In []: x = solve(A, b)

    In []: Ax = dot(A,x)

    In []: allclose(Ax, b)

    Out[]: True

\end{verbatim}

\section{Finding roots}

\begin{verbatim}

  In []: coeffs = [1, 6, 13]

  In []: roots(coeffs)

\end{verbatim}

Finding the roots of a function

\begin{verbatim}

In []: fsolve(sin(x)+cos(x)**2, 0)

\end{verbatim}

\section{ODE}

\begin{verbatim}

  In []: def epid(y, t):

  ....     k, L = 0.00003, 25000

  ....     return k*y*(L-y)

  ....

  In []: t = arange(0, 12, 0.2)

  In []: y = odeint(epid, 250, t)

  In []: plot(t, y)

\end{verbatim}

\end{document}