rule1=FourierCosTransform[e[x], x, p]-> eF[p,q,s];
rule2=FourierCosTransform[u[x], x, p]-> uF[p,q,s];
rule3=FourierSinTransform[[Theta][x], x, p]-> [Theta]F[p,q,s];
rule4=FourierSinTransform[v[x], x, p]-> vF[p,q,s];
rule={rule1,rule2,rule3,rule4}

eq=s (1 + s b) uF[x] + s (1 + s b) vF[x] + s [Theta]F[x] + s^2 b eF[x]

Distribute[FourierCosTransform[eq],x,p]