  In this paper, we derive two second- order of differential equation for the
gluon and singlet distribution functions by using the Laplace transform method.
We decoupled the solutions of the singlet and gluon distributions into the
initial conditions (function and derivative of the function) at the virtuality
$Q_{0}^{2}$ separately as these solutions are defined by: \begin{eqnarray}
F_{2}^{s}(x,Q^{2}) &=& \mathcal{F}(F_{s0}, \partial F_{s0})\nonumber
&&\mathrm{and} \nonumber
  G(x,Q^{2}) &=& \mathcal{G}(G_{0}, \partial G_{0}).\nonumber \end{eqnarray} We
compared our results with the MSTW parameterization and the experimental
measurements of $F_{2}^{p}(x,Q^{2})$.
