  The existence of a Picard-Vessiot extension for a homogeneous linear
differential equation has been established when the differential field over
which the equation is defined has an algebraically closed field of constants.
In this paper, we prove the existence of a real Picard-Vessiot extension for a
homogeneous linear differential equation defined over a real differential field
K with real closed field of constants. We give an adequate definition of the
differential Galois group of a Picard- Vessiot extension of a real differential
field with real closed field of constants and we prove a Galois correspondence
theorem for such a Picard-Vessiot extension.
