  Differential Equations are among the most important Mathematical tools used
in creating models in the science, engineering, economics, mathematics,
physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine,
environmental sciences, social sciences, banking and many other areas [7]. A
differential equation that has only one independent variable is called an
Ordinary Differential Equation (ODE), and all derivatives in it are taken with
respect to that variable. Most often, the variable is time, t; although, I will
use x in this paper as the independent variable. The differential equation
where the unknown function depends on two or more variables is referred to as
Partial Differential Equations (PDE). Ordinary differential equations can be
solved by a variety of methods, analytical and numerical. Although there are
many analytic methods for finding the solution of differential equations, there
exist quite a number of differential equations that cannot be solved
analytically [8]. This means that the solution cannot be expressed as the sum
of a finite number of elementary functions (polynomials, exponentials,
trigonometric, and hyperbolic functions). For simple differential equations, it
is possible to find closed form solutions [9]. But many differential equations
arising in applications are so complicated that it is sometimes impractical to
have solution formulas; or at least if a solution formula is available, it may
involve integrals that can be calculated only by using a numerical quadrature
formula. In either case, numerical methods provide a powerful alternative tool
for solving the differential equations under the prescribed initial condition
or conditions [9]. In this paper, I present the basic and commonly used
numerical and analytical methods of solving ordinary differential equations.
