  This paper is devoted to the study of the one dimensional non homogeneous
heat equation coupled to Dirichlet Boundary Conditions.
  We obtain the explicit expression of the solution of the linear equation by
means of a direct integral in an unbounded domain. The main novelty of this
expression relies in the fact that the solution is not given as a series of
infinity terms. On our expression the solution is given as a sum of two
integrals with a finite number of terms on the kernel.
  The main novelty is that, on the contrary to the classical method, where the
solutions are derived by a direct application of the separation of variables
method, on the basis of the spectral theory and the Fourier Series expansion,
the solution is obtained by means of the application of the Laplace Transform
with respect to the time variable. As a consequence, for any $t \ge 0$ fixed,
we must solve an Ordinary Differential Equation on the spatial variable,
coupled to Dirichlet Boundary conditions. The solution of such a problem is
given by the construction of the related Green's function.
