### Abstract:

In the first chapter of this thesis, several methods are used to
solve an n-th order linear ordinary differential equation with constant
coefficients together with n known initial values. The first
method is the standard elementary method where the general solution
of the differential system is found as a sum of two solutions u and
v where u is the solution of the homogeneous part of the ordinary
differential equation and v is any particular solution of the nonhomogeneous
differential equation. The method is not strong enough to
find a particular solution for every function that might be given as the
non-homogeneous term of the ordinary differential equation and so we
try a more powerful approach for finding v; hence the Lagrange's
method of variation of parameters. Following this, the method of
Laplace transforms is employed to solve the differential system.
In the second chapter the n-th order linear ordinary differential
equation is converted into a Volterra integral equation of second
kind and in the next chapter, the idea of the resolvent kernel of an
integral equation is introduced with some proofs of the existence and
convergence of the resolvent kernel of the integral equation. The
method of solving the Volterra integral equation by iteration is briefly
discussed.
The fourth chapter is devoted to solving the Volterra integral
equation with convolution type kernel by the method of E. T. Whittaker,
but the method is found to be very involved, and as a result, a method
suggested by G. C. Evans (1911) is employed in calculating the resolvent
kernels for kernels made up of sums of two exponential functions
(the method of iteration was applied to the same problem but it was
tedious--it took about 20 pages of writing) and finally the method provides
an easier way for calculating the resolvent kernel of the
Volterra integral equation corresponding to an n-th order linear
ordinary differential equation with constant coefficients.