  We investigate the most general exact solutions of Brans-Dicke cosmology by
choosing the scale factor "a" as the new independent variable. It is shown that
a set of three field equations can be reduced to a constraint equation and a
first order linear differential equation. Thus this new set of equations is
solvable when one supplies one of the following pairs of functions: ({\Phi}(a),
\r{ho}(a)), ({\Phi}(a), V(a)) or ({\Phi}(a), H(a)). A universe with a single
component energy-matter density is studied. It is seen that when there is no
constant energy density, the Hubble function still contains a constant term
which causes exponential expansion. This constant is expressed in terms of the
initial values of the universe. An early universe and the present universe with
dark energy are studied. In addition late time accelerated expansion is also
explained with cosmic domain walls. If we take Brans-Dicke parameter w>4*10^4
formulas of the Hubble function reduce to solutions of {\Lambda}CDM cosmology.
Therefore comparison of our results with recent observations of type Ia
supernovae indicates that eighty-nine percent of present universe may consist
of domain walls while rest is matter.
