Dear Bill,
Thank you for your answer. Unfortunately it would not work because the other functions, ga[t] and gb[t] have still as a parameter time, so replacing back by (t-u) won't solve the equation. I ran the code you very kindly provided and called the solution a new function RC, whatever the solution of my original differential equation should make the testRC=zero, and the solution that mathematica gives for the code you provide doesn't satisfy this :(
RC[t_] :=
RC[t] = ( (sd^2 + ra t) (sd^2 +
rb t) (sd^4 + ra rb t (u - C[1]) +
sd^2 (rb (t (1 - x) + x (u - C[1])) +
ra (u + x (t - u) + (-1 + x) C[1])))/(2 (ra -
rb)^2 sd^6 (t - u + C[1])));
testRC = - 2 RC[u - t] (gb[t]/sd^2 - ( x (gb[t] - ga[t]) )/sd^2) + ( (
x (x - 1))/(2 sd^2)) + 2 RC[u - t]^2 ((gb[t] - ga[t])^2/sd^2) +
D[RC[u - t], t];
Simplify[testRC]