x(t)= c1*e^t + c2*e^-t
Substitute in c1 and c2 gives:
x(t) = (1/2)(x0 + u0)(e^t) + (1/2)(x0-u0)(e^-t)
Then collect terms in x0 and u0:
x(t) = (1/2)*x0 (e^t +e^-t) + (1/2)*u0(e^t - e^-t)
EDIT: Sorry it's hard to read, it's just substitution and collecting terms.